Here's the resulting equation for $n = 3$, which is a little bit different than the traditional $3$-D cross product:

result[0] += v(0, 1) v(1, 2)
result[0] -= v(0, 2) v(1, 1)
result[1] += v(0, 0) v(1, 2)
result[1] -= v(0, 2) v(1, 0)
result[2] += v(0, 0) v(1, 1)
result[2] -= v(0, 1) v(1, 0)

For $n = 7…

I figured it all out. Thanks for your ideas!