I was told that any bilinear map $\displaystyle B: V\times V\longrightarrow \mathbb{F}$ (where $\displaystyle V$ is an n-dimensional vector space and $\displaystyle \mathbb{F}$ is a field of scalars) can be expressed as $\displaystyle B(u,v)=u^TAv$, where $\displaystyle A$ is an $\displaystyle n\times n$ matrix.

Does this statement require proof, and if so, how can I prove it?