How to prove the following:
- Prove that an eigenvalue of anti-symmetric matrix is either zero or imaginary.
- Prove that the eigenvectors of a symmetric tensor are orthogonal.
This answer is partly based on Hagen von Eitzen's comment.
Since $A^T = -A$, $(iA)^* = iA$; that is, $iA$ is self-adjoint, hence has real eigenvalues; it follows that $A$ has imaginary eigenvalues.
To prove the second point, you can generalize the approach for matrices given by Arturo Magidin here.