Let $V=F_5^2$ and let $Q:V \to F_5$ be given by $Q(x_1,x_2)=x_1^2-x_1x_2+2x_2^2$. I know that I can find the matrix representing the bilinear form $B$ corresponding to $Q$ by expressing $Q$ in the form $x^tAx$.
However, if I want the matrix representing $B$ in a specific basis, e.g. the standard basis, how do I do this?
EDIT:
Thinking about this some more, I guess I could use the matrix $A$ and then find a change of basis matrix $C$ and do it that way. But then how do I know what basis my original matrix $A$ was in to be able to do this?