Complemented real subspace

Question

Given the real vector space of the square matrices of order $3 (\mathscr M_3)$, let W be the subspace of the symmetric matrixes such that the sum of the elements of each of its rows is null. I have been able to find a basis in this subspace which is: $$W=\mathscr L\left[\left(\begin{matrix}1 &-1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}\ \right),\left(\begin{matrix}0 &0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{matrix}\ \right),\left(\begin{matrix}1 & 0 & -1 \\ 0 & 0 & 0 \\ -1 & 0 & 1 \end{matrix}\ \right)\right]$$ Now I want to find a complemented subspace, so its dimension has to be 6 and the intersection of both subspaces null, but I don't know which one it is. How can I find it? And is it a special type of real vector subspace (e.g. antisymmetric matrices...)?