Find a basis for the vector space of symmetric matrices with an order of $n \times n$ [duplicate]

Let $E_{ij}$ be the matrix with all its elements equal to zero except for the $(i,j)$-element which is equal to one.

Then a desired basis is $$ \frac{1}{2}\big(E_{ij}+E_{ji}\big), \quad 1\le i\le j\le n. $$

Hint: a symmetric matrix is determined by the coefficients on and above the diagonal. The matrix having $1$ at the place $(1,2)$ and $(2,1)$ and $0$ elsewhere is symmetric, for instance. Can you go on? Just take as model the standard basis for the space of all matrices (those with only one $1$ and all other entries $0$).

Let $\phi(A) = {A+A^T \over 2}$. Note that $\phi$ is a surjective map onto the space of symmetric matrices. Now choose a basis for the $n \times n$ matrices, then $\phi$ will map these into a spanning set. Now choose a maximal, linearly independent subset.

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