An explicit <basis> is
$$
\boxed{E_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad
E_2=\begin{pmatrix}0&0\\0&1\end{pmatrix},\qquad
E_3=\begin{pmatrix}0&1\\1&0\end{pmatrix}.}
$$
Indeed every real <symmetric matrix> $\begin{pmatrix}a&b\\b&c\end{pmatrix}$ has the unique <linear combination> $aE_1+cE_2+bE_3$.

Solution

Past exam of the mathematical methods I course of the University of Oxford / 2014 / 12 / ii / Solution

 Solution (source code)