Finding basis of 3x3 matrix

Lets assume we have a 3x3 matrix where the rows and columns sum to zero. Intuitively you know that for the rows to be 0, a + b + c = 0, or a + b = -c, so if you know the values of 2 elements, a and b, you know c.
$\begin{bmatrix} a & b & \underline{\hspace{.10in}} \\ c & d & \underline{\hspace{.10in}} \\ \underline{\hspace{.10in}}& \underline{\hspace{.10in}} & \underline{\hspace{.10in}} \end{bmatrix}$ so the _ represent values that are automatically determined one you give this matrix concrete values a,b,c,d which can be any real number. So what is the basis for the space of 2x2 matricies over real numbers?

$\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix},\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix},\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix},\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix}$
if you make each of these 2x2 matricies as the upper 2x2 matrix of a 3x3 matrix, it determines the last row and column

$\begin{bmatrix}1 & 0 & -1\\ 0 & 0 & 0 \\-1 & 0 & 1 \end{bmatrix},\begin{bmatrix}0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & -1 & 1\end{bmatrix},\begin{bmatrix}0 & 0 & 0 \\ 1 & 0 & -1 \\ -1 & 0 & 1 \end{bmatrix},\begin{bmatrix}0 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & 1 \end{bmatrix}$

thus giving you the basis.