A lot of things to do with gender *are* social constructs. Maybe there's a core that is not, and the rest is a web of expectations and associations that gets stuck to it.
But certainly, any individual trait (such as aggressiveness, or technical skill, or nurturing, or particular colors or kinds of clothing or manners of speaking) that people associate with gender, is only a vague statistical tendency and/or a cultural rule, and you can find exceptions and contradictions in other societies or in history.
I do not deny that gender (or in some cases, a lack thereof) is something we feel strongly at the core of our being. Nor do I deny that for some, there is an overwhelming feeling (or at least a vague feeling in my case) that their gender is not what society tells them it is.
But I believe it forms as we are very young and learning about the world through observing and imitating other people. Without society, I honestly don't think it makes sense to say gender exists.
You can't explain gender with hormones alone, either. I have a very testosterone-ish body. Male pattern baldness visible since age 18, lots of body hair, etc. But my personality isn't aggressive and "manly" and my identity is androgyne leaning toward feminine and simultaneously leaning toward agendered. My parents didn't raise me that way and society didn't lead me to it either, yet here I am.
Gender is an emergent property, an epiphenomenon. It comes from a synergy of things biological and psychological, and doesn't exist on its own without them. But what emerges from the input may be a surprise.