Answer
See below
Work Step by Step
Assume $A=\begin{bmatrix}
a & b & c \\ d & e &f \\g & h & i
\end{bmatrix} \in M_3(R)$
We can see that every matrix $A \in S$ is a scalar multiple of the matrix $\begin{bmatrix}
a & b &c \\ d & e & f \\ g & h & i
\end{bmatrix}=a\begin{bmatrix}
1 & 0 & 0\\0 & 0 & 0 \\ 0 & 0 & 0
\end{bmatrix}+b\begin{bmatrix}
0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 0
\end{bmatrix}+c\begin{bmatrix}
0 & 0 & 1\\ 0 & 0 & 0 \\ 1 & 0 & 0
\end{bmatrix}+d\begin{bmatrix}
0 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0
\end{bmatrix}+e\begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 1 \\ 0 & 1 & 0
\end{bmatrix}+f\begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 1
\end{bmatrix}$
Hence, the set $\{A=\begin{bmatrix}
a & b &c \\ d & e & f \\ g & h & i
\end{bmatrix}=\begin{bmatrix}
1 & 0 & 0\\0 & 0 & 0 \\ 0 & 0 & 0
\end{bmatrix},\begin{bmatrix}
0 & 1 & 0\\ 1 & 0 & 0 \\ 0 & 0 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0 & 1\\ 0 & 0 & 0 \\ 1 & 0 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 1 \\ 0 & 1 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 1
\end{bmatrix}\}$ spans $S$
