If it makes you feel any better, I have known plenty of aggressive and competitive women, especially at one facility I worked at where most of my coworkers were nurses. Some of them could get very nasty with one another at times. The last job I worked at, my boss was a woman, and would often tell us to pull our panties up and get on with the job, she was incredibly aggressive, competitive, and much more of a "man" than I ever was.
These ladies had estrogen coursing through their bloodstream for years, and it did not change them. I've always been on the passive side, hate competition, want to live in a world that is united in peace, etc, etc, and am not on HRT. I've seen them get in fights, stand up to others, where as I am the sort of person that will likely cry in direct confrontation, I really don't think it is a gender thing. It is just how gender roles are traditionally seen.
What I am saying is, you can be competitive and be a woman. I don't see any conflict here.