How to find a basis for this

We have a vectorspace (is it a vectorspace)$\displaystyle R^{2x2}$ we have a subspace $\displaystyle U \subseteq R^{2x2} $ where *U* is the amount of symmetric matrices, this means that the matrix $\displaystyle A {2x2}$

belongs to $\displaystyle U$ if and only if $\displaystyle A=A^T$

Then I have to find a basis. I would say that:

$\displaystyle E_1=\begin{bmatrix}1&0 \\ 0 & 0 \end{bmatrix} , E_2=\begin{bmatrix}0&1 \\ 0 & 0 \end{bmatrix} , E_3=\begin{bmatrix}0&0 \\ 1 & 0 \end{bmatrix} , E_4=\begin{bmatrix}0&0 \\ 0 & 1 \end{bmatrix}$

Is a basis.

Is that true and how can I show that?

Furthermore is it true that the dimension of $\displaystyle U$ is $\displaystyle 4$?