Given a plane, find another plane that is perpendicular (with the same origin). There are an infinite number of solutions because the plane could be rotated at any angle.
A simpler case is to simply find one vector that is perpendicular to an arbitrary vector.
When it comes to planes, you usually would be describing them as vectors anyway... specifically, each plane is described by a vector which projects orthonormally from its own surface. If the vectors projecting from two planes are perpendicular, then by definition so are the planes. There is a direct equivalence. In any case you would use the fact that two vectors are perpendicular if, and only if their dot-product is zero. As you said in your post, there are infinite solutions because of rotations. The only way to eliminate that is to impose another constraint. The rule is: for the number N dimensions you are working in, you need N+1 constraining parameters to be able to solve for the vector/plane you want. I can tell you a general approach, if you like, but it requires drawing for clarity. e-mail me if you are interested.