One way which I can think of doing this, is to consider instead the plane ; the angle between this plane and your vector will be the same. The idea is that now, this plane is a subspace of . Find a basis for it and orthonormalize it using the Gram-Schmidt process. Then calculate the projection of the vector on the plane ; the angle you seek will be the angle between and its projection on the plane, which you can find using the dot product.
Perhaps a bit messy but it shouldn't take more than 6 minutes if you are familiar with everything above. Otherwise there is probably another (shorter) way.