The generators of $SO(N)$ can be written as $(L_{ab})_{ij}=\delta_{ia}\delta_{jb}-\delta_{ja}\delta_{ib}$, with $1\leq i,j\leq N$ and $1\leq a<b\leq N$.
Obviously, these generators for a basis for skew-symmetric matrices. Putting the case $N=2$ aside, it seems to me that the matrices $M_{ab,cd}=L_{ab}L_{cd}+L_{cd}L_{ab}$ are more than enough to generate all the basis element of symmetric matrices. Combinations of $M_{ab,ba}$ should generate a basis for the diagonal, while $M_{ab,bc}$ with $a,c\neq b$ generate the off-diagonal symmetric basis (multiple times if $N>3$).
Does anyone have a reference for that, as it should be rather standard.