From Berkeley Problems in Mathematics, Summer 1981. Let $S$ denote the vector space of real $n\times n$ skew-symmetric matrices. For a nonsingular matrix $A$, compute the determinant of the linear map $T_{A}:S\rightarrow S, T_{A}(X)=AXA^{t}$
My main difficulty is I do not know how to associate $S$ and $T_{A}$ together. $S$ have $\frac{n^{2}-n}{2}$ dimension, and $T_{A}(X)_{ij}=\sum\sum\sum a_{ik}x_{km}a_{jm}$. Thus the entry for the linear transformation $T_{A}(X)_{ij}=\sum \sum a_{ik}a_{jm}$ looks nothing but the product of $\sum a_{ik}\sum a_{jm}$. I do not know how to solve it in better methods (find the eigenvalue was my first thought).