Well, if you look at the Fall, before God chases Adam and Eve out of the Garden He makes them some clothing. K'toneth - a "long, shirt-like garment." A dress, or a tunic. Something like the Greek chiton. Hmmm. The Bible also specifies that God is the same yesterday, today and tomorrow. He does not change. Hmmmm.
Joseph's 'coat of many colors?' Basically the aforementioned k'toneth. Tamar - King David's daughter wore a k'toneth. So. There must be something we're not seeing. 
The thing about cross-dressing was a common practice among the surrounding nations.
A key is the usage of kli gever - a "man's items." Swords. Armor. Kli is kinda-sorta something like "the appurtances of," or "the things of" and gever - can mean just a man, but is mostly used of a "strong man" - a warrior. In another attempt to identify the quintessential “men’s items,” Rabbi Eliezer ben Jacob, quoted in the Talmud (edited c. 800 C.E.), says, “What is the proof that a woman may not go forth with weapons to war?” He then cites our verse, which he reads this way: “A warrior’s gear may not be put on a woman” (B. Naz. 59a). He reads kli gever as the homograph kli gibbor, meaning a “warrior’s gear.”
This same understanding is followed by Midrash Mishlei (Proverbs) which contends that the Biblical character Yael in the Book of Judges kills General Sisera with a tent pin instead of a sword in order to comply with this law. It would have been “unlady-like” for her to use a sword — worse, a violation of the law — because a sword is a man’s tool and so the righteous woman of valor finds an alternate weapon. Note: the sword was lying right there, as the General likely took it off in order to go to sleep.
The current "majority opinion" under Jewish law is that: the scope of this law as interpreted by our sages is limited. It is designed especially to prohibit men and women from misrepresenting themselves as the other gender with the aim of illicit heterosexual activity. The mitzva does not prohibit cross dressing for the festival of Purim. It also would probably not prohibit cross-dressing in a private setting or for theatrical purposes, nor would it prohibit such dress when it would not mislead others as to the gender of the person under the clothes.
We also have to remember that things like the New King James, the NIV, the KJV and all the rest are NOT the Word of God. They are translations. Learn Biblical Hebrew and Koine Greek if you want to read the "real thing." The essentials are there and correct, but some of the fringes still carry the biases of those who translated it into English. If interested, look into the field of Textual Criticism.
Adam Clark's Commentary: "As the word...geber is here used, which properly signifies a strong man of war, it is very probable that armor is here intended; especially as we know that in the worship of Venus, to which that of Astarte or Ashtaroth among the Canaanites bore a striking resemblance, the women were accustomed to appear in armor before her.
Good day!
