Matrix representation of a bilinear form

Question

Let $a:\,\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a bilinear operator (i.e. $a$ is a linear operator in each component).

Then exists a square matrix $A\in \mathbb{R}^{n\times n}$ such that for all $u,v\in\mathbb{R}^n$ it holds

$a(u,v)=v^tAu$?

where $v^t$ is the transpose of the column vector $v$.

Answer 1

Hint: Suppose that $e_i$ is standard basis. Then what is $a(e_i,e_j)$?

Answer 2

Hint
Write $u = \sum_{i=1}^n u_i e_i, v = \sum_{i=1}^n v_i e_i$ and use bilinearity of $a$ on $$a(u,v)$$ Now compare this to $$v^T A u$$ This should give you an idea how $A$ looks like.