I have a question regarding this MO answer:
The answer says that in characteristic $2$, we cannot obtain a quadratic form from a bilinear form. I thought it was the other way around and now I am confused.
I thought that if $2 \neq 0$ then we have a bijection between quadratic $q$ and bilinear forms $b$, by getting $q(x) = b(x,x)$ from $b$ and $b(x,y) = \frac{1}{2}(q(x+y) - q(x) - q(y))$.
But if $2 = 0$, we cannot divide by $2$ hence we cannot go from quadratic forms to bilinear ones, but we can still go the other way.
Would you help me resolve my confusion? Thanks a lot.