Although I hate to say it, I must: Middle-earth is primarily a man's world. This is
not to say that there is an absence of strong female characters, because there certainly are Middle-earth women with strength and power. The most notable of these are Eowyn and Galadriel, but there are examples found in the Silmarillion, too. Still, it is the men who go off to war, not the women -- except Eowyn, of course, but if she had gone undisguised she certainly would have been turned straight around and sent back to her people.
I don't think there would be any competition in the Fellowship for a woman, pretty much for the reasons
lotrfreak90 gave. The only thing I disagree with in what he said is about Boromir, because it is stated that he is more interested in warfare than women.
Plus, if you think about it logically, when the Fellowship was created they had to select from the people they had in Rivendell. I don't think there were any female elven warriors around!
