Case 1) If A is symmetric => A = AT, where AT is transpose of A
So, (AT)² = (AT*AT) = (AA) = A²
Case 2) If A is skew-symmetric => A = -AT
So, (-AT)² = (-AT*-AT)=AA = A²
So, from both case 1 and 2 we can say that A² is symmetric if A is either symmetric or skew-symmetric.