Answer
$W$ is not a vector subspace of $M_{3,3}$.
Work Step by Step
Let $u=\left[\begin{array}{lll}{0} & {1} & {2} \\ {3} & {0} & {-1} \\ {0} & {2} & {1}\end{array}\right]\in W$ and $c=2\in R$. Now,
$$cu=2\left[\begin{array}{lll}{0} & {1} & {2} \\ {3} & {0} & {-1} \\ {0} & {2} & {1}\end{array}\right]=\left[\begin{array}{lll}{0} & {2} & {4} \\ {6} & {0} & {-2} \\ {0} & {4} & {2}\end{array}\right]$$
which is not an element of $W$. Hence, $W$ is not a vector subspace of $M_{3,3}$.
