@Pop By complementary reguli, I mean the set of three other skew lines which also lie on the cubic surface and pairwise intersect the original three skew lines. So, if I have 6 lines, each pair of intersecting lines must yield 7 independent conditions on the cubic form, thus three pairs impose 21 conditions (at most). Shouldn't I be able to determine the cubic this way?

@Wojowu Thank you! That's a useful link.