My teacher wrote:

(A+A^T)^T = A^T + (A^T)^T = A^T + A

While I get this algebraically, I don't see how it proves skew-symmetry. Like how does this relate to A = -A^T?! Also, I just wanted to confirm if the reason why the stuff (=bolded part) in (stuff)^T is A+A^T because we want to force them to be square matrices since you can only add matrices that have the same dimensions.

Any input would be greatly appreciated!

Thanks in advance!