In general, quadratic forms are not determined by its Stiefel-Whitney invariants, see Scharlau, Winfried: Quadratische Formen und Galois-Cohomologie. In: Invent. Math., 4 (1967), 238–264, p. 251 f.
However, if $\dim{q} \leq 3$, $q$ is determined by dimension, discriminant and Hasse invariant, see Lam, Tsit-Yuen: Introduction to Quadratic Forms Over Fields. Graduate Studies in Mathematics 67, American Mathematical Society 2005, p. 120, Theorem 3.21.