Answer
S does not span $R^3.$
Work Step by Step
Assume the combination
$$a(1,-2,0)+b(0,0,1)+c(-1,2,0)=(x,,y,z).$$
We have the system
\begin{align*}
a-c&=x\\
-2a+2c&=y\\
b&=z.
\end{align*}
The above system has a solution, if the determinant of the coefficient matrix is non zero but we have
$$\left| \begin{array} {cc} 1&0&-1\\-2&0&2\\0&1&0 \end{array} \right|= 0.$$
Consequently, the system has no solutions and so $S$ does not span $R^3$.