Let A be an nxn matrix. Assume that A is skew-symmetric. Then A^T = -A. The question is whether A^-1 is skew-symmetic. The answer is yes simply because of the fact that (A^-1)^T = (A^T)^-1. Note that
(A^-1)^T = (A^T)^-1 = (-A)^-1 = -A^-1
Let A be an nxn matrix. Assume that A is skew-symmetric. Then A^T = -A. The question is whether A^-1 is skew-symmetic. The answer is yes simply because of the fact that (A^-1)^T = (A^T)^-1. Note that
(A^-1)^T = (A^T)^-1 = (-A)^-1 = -A^-1