Dimension and base of a set

**Hi MHF**,

I'm unable to solve this problem. A little help would be welcome. (not a full answer if possible).

Find the dimension and a base of this set : $\displaystyle \{ A \in \mathbb{R}^{n\times n} \text {such that }A=A^T \}$.

My attempt : I'm tempted to answer $\displaystyle \dim S =n^2$ without thinking, but when I think I realize I don't know. $\displaystyle S$ is the set containing all the matrices nxn such that they are equal to their transposematrix. I'm not able to see further in order to find the dimension of $\displaystyle S$.

And once I know the dimension, I'll know how many vectors form a base of $\displaystyle S$.

**Thank you very much in advance**.