Trace of a power of a skew-symmetric matrix

Question

How to express ${\rm Tr}(A^n)$ (in terms of ${\rm det}\,A$), where $A$ is a skew-symmetric $m\times m$ matrix? With references if possible.

Answer 1

For even powers we cannot determine the trace from the determinant and order alone.

Let matrix $A$ be the $3×3$ matrix defined as follows:

$A_{1,2}=A_{2,3}=A_{3,1}=c,$

with other entries defined by skew-symmetry. Then $\det A=0$. But $A^2$ has all its diagonal elements equal to $-2c^2$, so its trace is $-6c^2$ which varies with $c$ despite the constant determinant.

The trace of an odd power of any skew-symmetric matrix is always zero. Can you see why?