is there a characterisation of the trace of a skew-matrix $M$ (say $n\times n$)? Clearly for $k=1$ we have $\bigwedge^kM=M$ hence the trace is zero, (since $M$ is skew).
What about $k=2$ and $k=3$? my intuition tells me that the trace of $\bigwedge^3M$ should vanish, but I can't relate it to the one of $M$.
The traces of $\bigwedge^k M$ are, up to sign, the coefficients in the characteristic equation of $M$, and so the elementary symmetric functions in the eigenvalues of $M$.
Over a field of characteristic not two, the non-zero eigenvalues of a skew-symmetric matrix $M$ always fall into pairs $\lambda$, $-\lambda$. Therefore for odd $k$ the trace of $\bigwedge^k M$ is always zero. For even $k$, the $k$-th elementary symmetric function of the eigenvalues is the $(k/2)$-th elementary symmetric function of the $-\lambda^2$ as above.