I can see that this is true but I am having trouble presenting a convincing argument.

Let V = $\displaystyle M_{n\times n} (\mathbb{R})$. Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.

The second part of the question however I have no idea where to start:

Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.