Suppose $a, b\in H^k(M, Z_2)$ are two k-cocycles on manifold $M$, valued in $Z_2$. We will assume $M$ to be orientable. For every such manifold, one can define the bilinear form $(a,b)= \int_M a\cup b$ which is defined only mod 2. Sometimes, one can also define the quadratic form of $(a,b)$, i.e. $q(a)\in Z_2=\{0,1\}$, satisfying $$q(a+b)= q(a)+q(b)+(a,b) \mod 2$$ My question is
What is the condition for $q(a)$ to exist?
In a special case $k=1$, I am aware that $q(a)$ exists if $w_2=0$, where $w_2$ is the second Stiefel-Whitney class of the manifold $M$. Then $q(a)$ depends on the spin structure $\sigma$, and sometimes one writes $q_{\sigma}(a)$ to high light the spin structure dependence. (Every orientable 2d manifold admits a spin structure. The emphasis here is that q depends on the additional data, i.e. the spin structure. )
For higher $k$, does the quadratic form exists only when $w_{k+1}=0$? Then what structure does $q$ depend on?