The most general bilinear form is
f(∑_{i}m_{i}e^{i},
∑_{i}n_{i}e^{i};
∑_{i}w_{i}e^{i})
= ∑_{i}φ_{i}e^{i}

where φ_{i} = ∑_{jkl}c_{i}^{jkl}m_{j}n_{k}w_{l}.

We seek the c's.
There are many but there is also much symmetry.

Here is how to recover a bilinear from b from a quadratic form q.
First if b(x, y) is bilinear then so is (b(x, y) + b(y, x))/2 which is symmetric and produces the same quadratic form.
We shall try for a symmetric bilinear form.
The function f above is not symmetric but we could symmetrize it.

q(x) = b(x, x)

q(x+y) = b(x+y, x+y) = b(x+y, x) + b(x+y, y)

= b(x, x) + b(x, y)+b(y, x) + b(y, y) = q(x) + q(y) + 2b(x, y)

b(x, y) = (q(x+y) −q(x) − q(y))/2

Voilà! and it is symmetric.