Answer
$$ - 1$$
Work Step by Step
$$\eqalign{
& \left| {\matrix{
0 & 0 & { - 1} \cr
{ - 1} & 0 & 1 \cr
0 & { - 1} & 0 \cr
} } \right| \cr
& {\rm{Calculating\, the \,determinant\, by \,expanding \,the \,third \,row}} \cr
& \left| {\matrix{
0 & 0 & { - 1} \cr
{ - 1} & 0 & 1 \cr
0 & { - 1} & 0 \cr
} } \right| = 0\left| {\matrix{
0 & { - 1} \cr
0 & 1 \cr
} } \right| - \left( { - 1} \right)\left| {\matrix{
0 & { - 1} \cr
{ - 1} & 1 \cr
} } \right| + 0\left| {\matrix{
0 & 0 \cr
{ - 1} & 0 \cr
} } \right| \cr
& {\rm{Solving}} \cr
& \left| {\matrix{
0 & 0 & { - 1} \cr
{ - 1} & 0 & 1 \cr
0 & { - 1} & 0 \cr
} } \right| = 0 + \left( {0 - 1} \right) + 0 \cr
& {\rm{Simplifying}} \cr
& \left| {\matrix{
0 & 0 & { - 1} \cr
{ - 1} & 0 & 1 \cr
0 & { - 1} & 0 \cr
} } \right| = - 1 \cr} $$
