If the matrix $A = \begin{bmatrix}0&x+y&1\\3&z&2\\x-y&-2&0\end{bmatrix}$ is skew-symmetric, then:

$x=-2, y=-1, z=0$

$A = \begin{bmatrix}0&x+y&1\\3&z&2\\x-y&-2&0\end{bmatrix}$

$A^T = \begin{bmatrix}0&3&x-y\\x+y&z&-2\\1&2&0\end{bmatrix}$

so skew symmetric

$⇒A=-A^T⇒\begin{bmatrix}0&x+y&1\\3&z&2\\x-y&-2&0\end{bmatrix}=\begin{bmatrix}0&-3&y-x\\-x-y&-z&2\\-1&-2&0\end{bmatrix}$

So,

$x+y=-3$  ...(i)

$x-y=-1$   ...(ii)

$z = -z$

$z=0$

Adding eq. (i) and (ii)

$2x = -4⇒x=-2$

form (i) $y = -1$