Vector Spaces & Bases · Challenge problem

Let $A = \begin{pmatrix}0 & 1 & 0 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\\0 & 0 & 1 & 0\end{pmatrix}$. The real $4\times 4$ matrices $X$ with $X^TA + AX = 0$ form a subspace of $M_4(\mathbb{R})$. What is its dimension?