Okay, wearing skirts in winter is
bullshit, yes. Get that changed if you can.
However: I fail to see how different uniforms encourages stereotyping and an assumption of behavior based on gender roles. That's something that, as has been said before, is based on the sex of the person in question rather than what clothing they wear, even less so if they're just wearing normal guy/girl clothes. Also, saying "girls wear skirts, guys wear trousers" doesn't encourage people to stereotype. What would encourage people to stereotype would be "girls wear low-cut tops and short skirts, guys wear wifebeaters and jeans fifteen sizes too large". That type of clothing has specific connotations and expectations attached to it--a skirt, in and of itself, does not.
Also, railing against gender role reinforcement as inherently bad--which sounds like what you're doing--is foolish. Gender roles are necessary in at least some extent to fit in to society--if you don't care about that or that's not important, then that's cool, but for most people, the ways you are expected to act as a man or a woman are important in interpersonal relations. However, in the real world, nobody much cares--it's about how you act as a person, as a professional and so on which transcends any concept of gender identity.
And while forcing or imposing gender identity on someone--forcing a guy to be all masculine and manly, forcing a girl to be all, well, girly--is generally a bad thing, I am still completely at a loss for how "girls wear skirts, guys were trousers" in any way does this. Yes, it treats the two sexes differently, but on a fairly petty level that I can't see as having a major impact on one's holistic gender identity. I had to wear a skirt a few times against my will in high school, and I never felt like my gender identity was being challenged or had undergone some radical change or whatnot. Similarly, I don't feel that wearing trousers has in any way influenced my male gender identity. Perhaps I'm weird and different, but I see no way that simply mandating a single piece of clothing that isn't something associating with any particular stereotype of either sex (what sort of guy wears trousers? All kinds. What kind of girl wears skirts? All kinds) is the degree of discrimination you make it out to be or could feasibly risk leading to any of the admittedly bad ends you have spelled out. If I'm missing something, I'd be more than happy to listen to an explanation, but so far all I've seen is "uniforms treat sexes differently, therefore it leads to an imposition of gender identity and a promotion of sex-based discrimination," which sounds to me to be about as silly as saying "mandating that doctors be held to a higher standard of medical knowledge promotes anti-intellectualism by segmenting the doctor population away from the rest of the general populace and setting them up as elite". Not quite the same sort of thing, but my point still stands. In debate terms (on the chance you do/have done CX debate), your harms do not follow from your status quo.
I feel I should clarify something here. I dislike the concept of a uniform. I do think it's stupid that girls have to wear skirts year-round no matter what, but that's for practical reasons (winter is goddamn cold sometimes). I don't think, however, that it's that big of a deal. By all means, try to get your uniform policy changed, I can't stop you (as I live in the States). I just think you're going about it for completely the wrong reason and using odd justifications that don't make much sense. There are better reasons to get rid of elements of dress code. It removes one avenue for possible self-expression within socially acceptable bounds (i.e. you couldn't wear a shirt saying "Fuck the Jews, thank god for the Holocaust" and expect to get away with it) is the biggest one. But arguing that skirts versus trousers leads to a division of the school body and a promotion of gender stereotyping via enforcement of gender roles is petty and lacking perspective. Or says a lot about the shallowness of the student body at your school if they legitimately divide themselves into camps solely because "they wear skirts and we wear trousers" (and vice versa).