Find basis for $V = \{ A \in M_{nn} | A$ is symmetric$\}$ The full problem statement is: 
$A \in M_{nn}$ is called symmetric if $A^T = A$ and $B^T=B$. Let $V = \{ A \in M_{nn} | A$ is symmetric$\}$.  Determine a basis for $V$.
I am having difficult starting with this problem.  If I understand correctly, $V$ is a set of symmetric matrices.  Meaning the only restriction is that for all matrices in $V$, every elements $e_{ij}$ for $0 < i \le j \le n$ must equal their counterparts $e_{ji}$.
The identity matrix $I_n$ seems like it is a basis but I don't know how to determine that.  I could verify that $I_n$ spans $V$ and is linearly independent proving it is a basis of some kind, but not necessarily of $V$.
I feel like I'm close to the solution but am missing something.

Note that the matrices $E_{ij}$ where the only non zero entry is $a_{ij}=1$ form a basis of $M_{nn}$. Now, $E_{ii}'s$ are present in your basis for $i=1..n$. The other basis members are $(E_{ij}+E_{ji})/2$. 
