Symmetric and skew-symmetric matrices

Subspaces. $S$ and $A$ are defined by linear conditions ($M^T-M=0$ and $M^T+M=0$), hence are kernels of linear maps, hence subspaces.

Intersection. If $M\in S\cap A$, then $M^T=M$ and $M^T=-M$, so $M=-M$, i.e. $M=0$.

Decomposition. For any $M$:
$$M = \underbrace{\frac{M+M^T}{2}}_{\in S} + \underbrace{\frac{M-M^T}{2}}_{\in A}.$$
Indeed, the first term is symmetric and the second is skew-symmetric.

Conclusion. $\mathcal{M}_n(\mathbb{R})=S\oplus A$. Their dimensions are $\frac{n(n+1)}{2}$ and $\frac{n(n-1)}{2}$, summing to $n^2$.