This is pretty standard. Do you know the formula for the cross product of two vectors?
Let a = <1, 1, -1>, b = <1, -1, 1>
Then,
a x b = [[0, -a_3, a_2], [a_3, 0, -a_1], [-a_2, a_1, 0]]*[[b_1], [b_2], [b_3]], where the * is matrix multiplication.
And thus we have:
a x b = [[0, 1, 1], [-1, 0, -1], [-1, 1, 0]]*[[1], [-1], [1]]
= <0, -2, 0>
You can also compute the cross product using determinants, which some people find easier. I personally find this much faster.