For a bilinear form on $ \mathbb R^2$ the matrix of the bilinear form, A, with respect to the standard basis is $a_{ij}= \langle e_i, e_j\rangle$, for $ i =1,2$, j = $1,2$. Then for any two vectors $x,y$ one can write $\langle x,y\rangle = x^tAy$.
i don't understand the matrix of a bilinear form in say $ \mathbb R^{2*2}$. If I take a bilinear form $\langle A,B\rangle = trace(AB)$ with respect to the standard basis $e_{ij}$ I get a $4*4$ matrix, C, of the form. Then the product $\langle A,B\rangle$ = $A^tCB$ isn't defined since is consists of a $2*2$ matrix multiplied either side of a $4*4$. At least this is my incorrect intuition. What is the correct way to view the matrix of a form on a space of matrices?