Reference pn bilinear forms

Question

For $V$ a $\mathbb{k}$-vector space, with a choosen basis, bilinear forms $\sigma:V \times V \to \mathbb{k}$ correspond to matrices $S \in M_n(\mathbb{k})$, where $n$ is the dimension of $V$. The properties of the bilinear form correspond to properties of the matrix, for example non-degeneracy of $\sigma$ corresponds to $S$ having non-zero determinant. Where can one find a list of such correspondences along with proofs?