Minimal polynomial of $T(A) = A^\top - A$

I do not know if i can say anything better than what you have done...

You have seen what $T^2$ would be... this is what you actually have to do..

see what would $T,T^2,T^3\cdots$ be and check for a liner combination that would result zero map ..

You have seen the very first non trivial power of $T$ namely $T^2$ and realized it as $-2T$

So, You have $T^2=-2T$ and remaining thing i want to say is not any better than yours..

So, What you have done is natural for me..

P.S : All this is just for your statement I guessed $p(t)$ and I'm not sure on how to actually find the polynomial