Invertible skew-symmetric matrix I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the determinant $0$ and therefore the matrix is not invertible?

The diagonal of a skew-symmetric matrix is always $0$ does not mean that its determinant be $0$. Look at following example:
$det\left[ \begin{array}{}
   0 & 1 \\
   -1 & 0 \\
  \end{array}  \right]=1
$
Its inverse is:
$\left[ \begin{array}{}
   0 & -1 \\
   1 & 0 \\
  \end{array}  \right]
$
