find the kernel of the linear transformation on $M_n(\mathbb{R})$ $L: M_n(\mathbb{R}) \rightarrow M_n(\mathbb{R})$ is defined by $L(A) = A + A^T$ 
I found the kernel of this linear transformation and it will consist of matrices of the following form: 
$A$ such that $a_{ii} = 0$ for all $i \in \{1, \cdots, n\}$ and $a_{ij} = -a_{ji}$ for all $i \neq j$.
But what is the basis for this and the dimension?    

The simplest basis I can think of.
For each $i,j$ pair with $i>j, a_{ij} = 1$ and $a_{ji} = -1$  Place zeros elsewhere.
How many vectors can be define this way? It is the equal to the number of elements in the upper triangle excluding the main diagonal.
$\frac 12 (n)(n-1)$
