Signature of a bilinear form in the space of $2 \times 2$ Hermitian matrices

Question

We have a bilinear form,

$$\langle A, A' \rangle = \det(A+A') - \det A - \det A'$$

on the real vector space of Hermitian $2\times 2$ matrices.

If I have a basis of the space, I would calculate the norms of the basis vectors, but here we don't. What should I do?