Q:

Find a Basis for the subspace S of Vector Space V

What is the method for finding a general solution? 
For example. 
Let S be the subspace of V, where V = all [2x2] symmetric matrices.  
So A transpose(A^t) = A.

What is a Basis for subspace S?  I can't find an examples of this.  I have been trying to use properties of Linearly Independant Combo's and Vector Spaces but I'm coming up with a whole lot of nonsense. 

A:

$\text{Hint:}\\
        \begin{pmatrix}
        a_{11} & a_{12} &...& a_{1n} \\
        a_{12} & a_{22} &...& a_{2n} \\
...&...&...&...\\
        a_{1n} & a_{2n} &...& a_{nn} \\
        \end{pmatrix}
=\left\{a_{11}\begin{pmatrix}
        1 & 0 &...& 0 \\
        0 & 0 &...& 0 \\
...&...&...&...\\
        0&0&...&0 \\
        \end{pmatrix}+a_{22}\begin{pmatrix}
        0 & 0 &...& 0 \\
        0 & 1 &...& 0 \\
...&...&...&...\\
        0&0&...&0 \\
        \end{pmatrix}+...+a_{nn}\begin{pmatrix}
        0 & 0 &...& 0 \\
        0 & 0 &...& 0 \\
...&...&...&...\\
        0&0&...&1 \\
        \end{pmatrix}\right\}+\left\{a_{12}\begin{pmatrix}
        0 & 1 &...& 0 \\
        1 & 0 &...& 0 \\
...&...&...&...\\
        0&0&...&0 \\
        \end{pmatrix}+...+a_{1n}\begin{pmatrix}
        0 & 0 &...& 1 \\
        0 & 0 &...& 0 \\
...&...&...&...\\
        1&0&...&0 \\
        \end{pmatrix}\right\}$

