Sets with many pairs of orthogonal vectors over finite fields

Let n be a positive integer and B be a non-degenerate symmetric bilinear form over Fq , where q is an odd prime power and Fq is the finite field with q elements. We determine the largest possible size of a subset S of Fq such that |{B(x,y) |x,y ∈ S and x 6= y}| = 1. We also pose some conjectures concerning nearly orthogonal subsets of Fq where a nearly orthogonal subset T of Fq is a set of vectors in which among any three distinct vectors there are two vectors x,y so that B(x,y) = 0.