Let $H$ be the subset of all 3x3 matrices that satisfy $A^T$ = $-A$. Carefully prove that $H$ is a subspace of $M_{3x3} $ . Then find a basis for $H$.

Let $A_1, A_2 \in H$. To show that $H$ is a subspace, we need to show that 
\[\alpha A_1 +\beta A_2 \in H,     \text{for all}   \alpha,\beta \in \mathbb{R}.\]
We have 
\[(\alpha A_1 +\beta A_2 )^T=\alpha A_1^T+\beta A_2^T=\alpha (-A_1)+\beta (-A_2)\]
\[=-(\alpha A_1 +\beta A_2).\]
Hence
\[(\alpha A_1 +\beta A_2 )^T=-(\alpha A_1 +\beta A_2),\]
which means $\alpha A_1 +\beta A_2 \in H$, $\alpha,\beta \in \mathbb{R}.$ So $H$ is a subspace of $M_{3 \times 3}$.