possible signatures of bilinear form on subspaces

Question

Suppose to have been given the signature of a symmetric bilinear form on a finite dimensional vector space. Is there a general rule to get all the possible signatures of the restriction to subspaces of codimension 1? For instance, I know that if the signature is (-,+,+,+) all subspaces of codimension one have signatures (+,+,+), (-,+,+) and (0,+,+). If I try with (-,-,+,+) I would say that the answer is (0,-,+), (-,-,+), (-,+,+), but I do not have a rigorous argument and furthermore I do not trust my intuition for bilinear forms in vector spaces of larger dimension, e.g. (-,-,0,+,+,+,+). Is there some result which allows us to write down all the possible signatures? What about larger codimension?