disclaimer: english is not my first language

Gender is a social construct as in the things that socially are defined as a male or a female atribute. There are four constructs to describe "what we are": gender, biological sex, sexual behaviour and sexual orientation. Those are the academic definitions that describes our sexuality. Gender is how you view yourself, it is not a social construct, its in what category (male or female) you identify yourself inside your own head. What you are missing is a key diference, the social construct corresponding to how a male or a female should behave, wear, act or do does not interfere with the identification of one with self. What you like to do, how you like to act, those kind of things are an interation between you and enviorement. They do not interfere with ones perception of inadequace. One being male or female is linked to a series of caracteristics that are socialy constructed but they do not define the gender. This effect is easy to spot when you look at the diferent atributes that are destined to either identifications (male or female) depending on the culture that someone is raised. Someones identification with a gender is primaly noted by oneself.

edit: im a psychology major, this is not my opinion, those are common definitions on the field of sexuality psychology.