Symmetric $2 \times 2$ matrices have form $\begin{pmatrix} a & b \ b & c \end{pmatrix}$. Basis: $\begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}$, $\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$, $\begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}$. Dimension is 3.