Angle between matrices This is the problem from my homework:

If $A$ is antisymmetric matrix, and $S$ is symmetric matrix where $A,S \in M_n (\mathbb{R})$, determine the angle between them according to the inner product defined as $(A,B)=\text{tr}(AB^T)$.

I have tried calculating the angle between two pairs of concrete matrices from space $M_2(\mathbb{R})$, and the angles don't turn out to be the same. Am I doing something wrong? If it's even possible, how to determine that angle? Thanks.

The trace verifies the property $tr(A B^T)=tr(B A^T)$, so$$\cos(\theta)=\frac{tr(A B^T)}{\sqrt{tr(A A^T)tr(B B^T)}}=\frac{tr(B A^T)}{\sqrt{tr(A A^T)tr(B B^T)}}$$
