matrix eigenvalue problem for a skew symmetric matrix of order 5

Question

Let $-2+5i,\ 3i$ be two eigenvalues of a $5\times5$ skew symmetric matrix $A$. Find the other eigenvalues of $A$.

In a skew symmetric matrix the eigenvalues can be $0$ or pure imaginary.So I think the other 3 eigenvalues are $2-5i,-3i$ and $0$. But only orthogonal matrices can have its eigenvalues existing as complex conjugates in pair. Can someone help with the answer..

Answer 1

"In a skew symmetric matrix the eigenvalues can be 0 or pure imaginary."

I think this is only valid for real valued matrices.

Answer 2

Eigen values of a skew symmetric matrix can be 0 or pure imaginary.Also trace of Skew symmetric matrix=0 and det(skew symmetric matrix)=0. This is possible only if the the other 3 eigen values are 0 ,-2-5i and -3i.