While the Israelites mirrored the communities around them in being a patriarchal culture, that was never of God's command.  There are also a few matriarchal cultures mentioned in the Bible, and they are not condemned.  The Queen of Sheba, who visited Solomon was from a matriarchal community.  The Kandake who had the Ethiopian eunuch in the book of Acts was from a matriarchal culture.

Patriarchal culture aside, women served in the temple in the Old Testament.  We read of women serving at the temple/tabernacle entrance in Exodus 38:8 and 1 Samuel 2:22.  Women were singers and musicians, such as we see in such places as Ezra 2:65 and Nehemiah 7:67.   Women even helped rebuild the Wall in Jerusalem, as we see in Nehemiah 3:12.

While the culture was patriarchal, it was not commanded of God to be so.  In fact, Genesis 1:28 shows that God's design was a biarchy, of man and woman equally ruling together. Further, God's design was that the husband leave his family to join his wife (Gen. 2:24, and reiterated by Jesus in the Gospels) which is counter-patriarchal.  While patriarchalism demands patriarchal succession, it was not a command. In fact, Job made both his male and female children equal heirs, as we see in Job 42:15.  Also in Job chapter 42, we read that only his daughters were mentioned by name, and not his sons.  

There were also many women leaders in the Old Testament, such as Huldah, Deborah, Jael, and Miriam.

We see no shortage of powerful women of God in the New Testament, as we have a specific female apostle mentioned by name in Romans 16:7.  Her name was Junia.  The New Testament also mentioned many female disciples who followed Jesus. Here is a list of some of those women mentioned: Mary Magdalene, Joanna, Susanna, Mary - Jesus's mother, Martha and her sister Mary, Mary - the mother of James and Joseph, Mary - the mother of the sons of Zebedee, Salome, and "many others," as the Bible states.

In Mark 14, we see a woman in the role of a priest, in which she anoints Jesus not only for his impending death and burial, but this is also how kings were anointed in the Old Testament by priests. Thus emphasizing "Christ," which is the Greek "Christos," which literally means "the anointed One." By a woman, Jesus was anointed to be in the order of Melchizadek, as the King and Priest. This is written about in the book of Hebrews, which, by the way, was likely written by a woman.  More on that shortly...

There is no shortage of female church leaders mentioned in the Bible, and many are mentioned by name.  In fact, the first church preacher mentioned by name in the New Testament is female.  Her name is Phoebe:

"And I commend you to Phoebe our sister -- being a minister of the church that [is] in Cenchrea --that you may receive her in the Lord, as does become saints, and may assist her in whatever matter she may have need of you -- for she also became a leader of many, and of myself." -Romans 16:1-2 Revised Young's Literal Translation

The first Christian church on the European continent had a female leader.  Her name was Lydia.  We read about her in Acts chapter 16.

Other female church leaders in the New Testament are: Priscilla, Anna, Euodia, Syntyche, Chloe, Nympha, Mary - the mother of John Mark, and many others. 

With so much female leadership in both the Old and New Testaments, why would one assume that none of the Bible was written by females?  There is NO evidence that the Bible was written by only men.  In God's eyes, we are all the same, regardless of gender, race, or social status, as we see clearly taught in Galatians 3:28.

If we look through the annals of history, we find out that the very first published author was a woman. Her name was Enheduanna.  She was the daughter of Sargon, the Akkadian king.  

It is argued that due to the cultures in and around Israel being patriarchal, women were not generally taught how to read.  This did not stop some women from teaching themselves how to read.  Even when women were illiterate, we see that men wrote down these women's words, and put them in certain places in the Bible.  This would count as women authorship in such situations, just like some of the Pauline Epistles (if not all of them) may have been dictated by Paul, and written down by someone else.

Judges chapter 5 was written by Deborah, either directly, or listened to by someone else, and they wrote down her words.

Proverbs chapter 31 was authored by a woman, where King Lemuel recorded his mother's words.

Hannah authored 1 Samuel chapter 2, verses 1-10, where it records her prayer.  This was either directly written by her, or her words were listened to, and recorded by someone else.

 The words of the prophet Huldah are recorded in 2 Kings 22:16-20.

The mother of Jesus - Mary's words are recorded in Luke chapter 2, verses 46-55.

What about book authorship in the Bible?   Before we get into that, I need to take you on a side subject, to show you a prevalent mindset, which will be entirely relevant to this topic...

Take a look at the below verses, and see if something jumps out at you, that you may or may not have ever noticed before. I will underline the parts I want you to notice:

"I am the door: by me if any man enter in, he shall be saved, and shall go in and out, and find pasture."  -John 10:9 KJV

"For this is good and acceptable in the sight of God our Saviour; Who will have all men to be saved, and to come unto the knowledge of the truth."  -1 Timothy 2:3-4 KJV

"He who believes in Him is not condemned; but he who does not believe is condemned already, because he has not believed in the name of the only begotten Son of God."  - John 3:18 NKJV

"For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation."  Romans 10:10 KJV

"The Spirit and the bride say, "Come!" Whoever is thirsty, let him come; and whoever wishes, let him take the free gift of the water of life." Revelation 22:17  NIV 1984

"Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me." John 14:6 KJV

"Not everyone who says to me, 'Lord, Lord,' will enter the kingdom of heaven, but only he who does the will of my Father who is in heaven." Matthew 7:21 NIV 1984

"And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.” Acts 4:12 ESV

"Whoever believes in the Son has eternal life; whoever does not obey the Son shall not see life, but the wrath of God remains on him." John 3:36 ESV

"No one can come to me unless the Father who sent me draws him. And I will raise him up on the last day." John 6:44 ESV

"For the grace of God that brings salvation has appeared to all men." Titus 2:11 NIV 1984

"For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast." Ephesians 2:8-9 KJV 

"And they were exceedingly astonished, and said to him, “Then who can be saved?” Jesus looked at them and said, “With man it is impossible, but not with God. For all things are possible with God.”  Mark 10:26-27 ESV

"But he who stands firm to the end will be saved."  Matthew 24:13 NIV 1984 

"For whoever would save his life will lose it, but whoever loses his life for my sake will find it." Matthew 16:25 ESV

"Be watchful, stand firm in the faith, act like men, be strong."  1 Corinthians 16:13 ESV

"I appeal to you therefore, brothers, by the mercies of God, to present your bodies as a living sacrifice, holy and acceptable to God, which is your spiritual worship." Romans 12:1 ESV

What did you notice?  To a person new to the faith, who has had no prior teaching in Christianity, they are going to read these and the MANY other Scriptures like this that I didn't quote above, and they are going to conclude that only men go to heaven, and Christianity is only for males.  Farfetched?  No.  Multiple female children, ranging in ages 5-12 have been shown some of these Scriptures in Sunday School, and when asked, they state that these Scriptures are only for boys.

If you look up all of the above in the Greek, they are ALL gender neutral!  Even in the case of "brothers" above, the Greek word adelphoi means "brothers" or "siblings" in a physical family, but when addressing a crowd, it means "brothers and sisters."

In the nineteenth century, the church of England had to convene to discuss many church issues, and one of them was that many people thought that women couldn't go to heaven, because of the above verses and more.  The council decided to teach that "he," "man/men," etc. was often used in the gender neutral sense. This did little to help the people, because there are places in the Bible where it is meant as male - he, and male people - men/man.  The problem then became, how do we know when the Bible is talking about all people, versus just talking about males?

You see, EVERY instance of male pronoun or word in all of the above Scriptures I just shared with you is not male in the Greek, but is strictly gender neutral.  I quoted from both older and newer Bible translations, and we see that even though the Greek is gender neutral in those passages, the translators still translated in the male.  Why?  One reason is that two of the above translations had all male, outwardly patriarchal translators.  They translated from their worldview. 

(Note: If you are looking for more gender accurate bibles, I suggest the New Revised Standard Version updated edition, and the Common English Bible.)

Many men don't notice these above passages like women do when reading the Bible, because males are included in these translations.   Women are not.  Women are not - "he," "man," "men," and "brother."  We do not feel included.  In fact, many of us shivered in FEAR when we read the above passages for the first time, because we thought that maybe women can't go to heaven, because Eve ate the forbidden fruit first.  In case you don't know, that is a teaching some do actually hold to.

My point in this side topic is to show you that a patriarchal mindset will lead to patriarchal translations.  When men wrote some of the books of the Bible, they did so from a patriarchal mindset, because that's what they were raised in and knew.  Therefore, when we come across a book in the Bible about a woman, or chapters in the Bible about women, it is not a reach to consider that that part was actually authored by a female. 

And this segues into possible female authorship of some of the books of the Bible...

What about the book of Ruth?  The main people in that book are women - mostly Ruth and Naomi.  A man writing the contents of the book of Ruth would have been unheard of, because with the culture being patriarchal, men automatically wouldn't have given much thought to women, just as the male, patriarchal translators mentioned above seemed to give no thought to how their sexist translations are excluding women, and making many females fear that they are going to hell, purely because they don't possess a penis.

This is the same case for the book of Esther.  The main person and hero in that book is the woman Esther.  

There is literally NO evidence that Ruth and Esther were written by men.  In fact, logic and culture leans more to the side of both of these book having been authored by females.

Another Old Testament book in dispute over authorship possibly being female is Ecclesiastes.  Ecclesiastes opens by saying 'The words of the preacher/teacher.'  The Hebrew word for 'preacher' or 'teacher' in this verse is qōheleṯ.  According to Strong's, 'qōheleṯ is the feminine of the active participle from qāhal, a female lecturer or preacher.'

What about the New Testament?  In Acts chapter 18, we are introduced to Priscilla and her husband Aquila.  Here we see that counter to culture, this married couple appeared to be egalitarian, which helped usher in Galatians 3:28:

"In Christ, there is no difference between Jew and Greek, slave and free person, male and female. You are all the same in Christ Jesus." -Galatians 3:28 NCV

In Acts 18, we read that Priscilla and Aquila teach Apollos, a Jew, more fully about Christ. Notice it mentions Priscilla before her husband Aquila there.  This is the doctrine of Order of Prominence, in which when two or more people or groups are mentioned, the more prominent one is mentioned first.  In regards to tent-making, we see Aquila mentioned first.  When it comes to teaching Jews about how Christ is the fulfillment of the law, etc - we see Priscilla mentioned first.  We also see in reference to their church they later started, that Priscilla/Prisca (see Romans 16) was the main leader of that church, as her name was mentioned before Aquila's.  

Acts 18 shows a woman teaching a man in Ephesus.  Sound familiar?  1 Timothy 2:12 was in Ephesus, and many people quote the traditional and incorrect patriarchal translation of 1 Timothy 2:12, and they say this verse says women aren't allowed to teach men.  That can't be what 1 Timothy 2:12 means, or we have a contradiction in the Scriptures.

Like the many verses I showed you being wrongly translated with a patriarchal worldview, the same is the case with 1 Timothy 2:12.  For example, take a look at it in other translations:

"I don’t allow a wife to teach or to control her husband. Instead, she should be a quiet listener." CEB

"Moreover, in the area of teaching, I am not allowing a woman to instigate conflict toward a man. Instead, she is to remain calm."  ISV

"I don’t advocate that the newly converted women be the teachers in the church, assuming authority over the men, but to live in peace." TPT 

Why are we learning about Priscilla?  I bring her up, because she is likely the author of the book of Hebrews.

- As we saw, Priscilla taught Apollos, who was a Jew well-versed in the Scriptures.  We can therefore derive that Priscilla must have been even more adept in the Hebrew Old Testament, and in showing Christ's fulfillment of  it, than Apollos was.  She was an expert in this.  What's the main theme of the book of Hebrews?  It happens to be Priscilla's expertise - Christ fulfilling the Hebrew Old Testament Scriptures.

- The fact that the book of Hebrews is anonymous also leans towards female authorship.  Through the centuries, many books were written by females, and were either left anonymous, or with male pseudonyms.  A classic example of this are the Bronte sisters, who originally released their books under male pseudonyms.  Yet if you read their books, you can see it was written by females, just as we can see that similarity in the books of Ruth and Esther in the Bible.

- Hebrews chapter 11 is written in a feminine perspective, as can be seen from verses 11, 23, 24, 31, and 35. 

- Hebrews 6:1-3 and 9:8-10 pair with the teaching Priscilla gave to Apollos in Acts 18.

- Hebrews 13:24 says that the writer is from Italy, and Acts 18:2 shows Priscilla is from Italy/Rome.  Further, Priscilla/Prisca lead a church in Rome, as we see in Romans 16:3-5a.

There has only been one argument used to state that Hebrews must have been written by a man, and that is, as stated in the notes of the NIV Study Bible on Hebrews 11:32:  "to tell. Translates the masculine form of a Greek verb, indicating that the author of Hebrews was a man."

This is misleading and incorrect.  First of all, Greek verbs are not masculine or feminine, but some Greek nouns are masculine or feminine.  Some who are more adept with this fallacious argument will state that the participle to the verb "to tell" is in the masculine.  This, however is a half-truth, and therefore falls under the category of a white lie.  That particular participle can be used in the masculine, OR in the neuter.  If the author were female, she was clearly using it in the neuter sense.

In concluding, we see that the likelihood of some female authorship in the Bible is not only viable, but actual.