Recovering a quadratic form from its orthogonal group

We prove the following statement: Let K be a field of characteristic different from 2, let V be a K-vector space of finite dimension n > 1, let q and r be two quadratic forms on V, let O(q) and O(r) be the respective orthogonal groups, and let Q, R : V × V → K be the respective associated symmetric bilinear forms. Then we have O(q) = O(r) ⇔ Kq = Kr.