symmetric bilinear form on $\mathbb{Z}_2$

Question

Find all symmetric bilinear forms of a vector space $V$ of finite dimension on $\mathbb{Z}_2$.

As every bilinear form is represented by a matrix then the idea is to find the set of matrices representing these transformations, but not how to proceed, I appreciate any hint.

Answer 1

Hint:

Take $V = (\Bbb Z_2)^n$. Let $M \in \Bbb {Z_2}^{n \times n}$. Let $e_1,\dots,e_n$ denote the canonical basis vectors. We note that for $M$ to be a symmetric bilinear form, we must have $$ (e_i)^TMe_j = (e_j)^T M e_i $$ Why is this condition also sufficient?