Gender is constructed, but not by an 'I' or a 'we' who stands before that construction in any spatial or temporal sense of 'before.' the 'I' neither precedes nor follows the process of this gendering, but emerges only within the matrix of gender relations themselves.
Gender is a stylized repetition of acts, a constructed identity, a performative accomplishment which the mundane social audience, including the actors themselves, come to believe and to perform in the mode of belief.

Gender is constructed by one who takes on or appropriates that gender and could in principle, take on some other gender.

Turtles give away their gender by the noise they make. Males grunt and females hiss
Oysters are able to change from one gender to another and back again
Gender is a deeply grammatical concern (such that rather than gender, the question of 'what 'gender' is grammatically about'becomes the central category of feminist critique.
I'm straighter than the pole your mother dances on
Gender is a system of meanings, of ways of thinking, images and words that first shape how we experience, understand, and represent ourselves as men and women, but that also do more than that... they shape many other aspects of our lives and culture. In this symbolic system, human characteristics are dichotomized.
Earthworms are hermaphroditic, containing both male and female organs
The male firefly's light is twice as bright as a female's
Gender is achieved through repeated acts, where the acts themselves actually constitute the gendered subject instead of expressing it, hence, 'choice' violates the very concept such that one cannot, say, decide to be a man and subvert one's identity.
Gender is a narrative that is sustained by the tacit collective agreement to perform, produce, and sustain discrete and polar genders as cultural fictions is obscured by the credibility of those productions and the punishments that attend not agreeing to believe in them.
Gender is a social artifice radically independent of sex. Our ideas of what women and men are reflect nothing that exist eternally in nature. Instead they derive from customs that embed social relations of power.