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.

Hint: As $A$ is antisymmetric, $S$ symmetric, we have 
$$ SA^t = -SA, \quad AS^t = AS $$
Hence
$$\def\tr{\mathop{\rm tr}}\tr(SA^t) = -\tr(SA) = -\tr(AS) = -\tr(AS^t) $$
