In this generality, it is not at all clear what you mean by 'any general rule'. Of course, there *is* a general rule: Compute the rank of the appropriate matrix or linear map. However, computing that rank can easily be nontrivial, and there is no universal way to simplify this problem.

In your specific case, of the system of equations on the $\partial_kb_{ij}$, the number of independent equations *is* $C(n,3)$ after all, because you are asking for the rank the exterior multiplication mapping
$$
V\otimes\Lambda^2(V)\to\Lambda^3(V).
$$
This mapping is, of course, onto (because the map isn't zero and the right hand side is an irreducible $\mathrm{GL}(V)$-module), so the rank is the dimension of $\Lambda^3(V)$, which is $C(\dim V, 3)$.

As an example, though, of a case in which you don't necessarily know the correct answer right away, consider the case of the Bianchi identities
$$
R_{ijkl}+R_{iklj}+R_{iljk} = 0
$$
where $R_{ijkl}=-R_{jikl}=-R_{ijlk}$. It is less obvious how many independent equations this is because you may not know the range of the corresponding map
$$
\Lambda^2(V)\otimes\Lambda^2(V)\to V\otimes\Lambda^3(V)
$$
because the right hand side isn't irreducible as a $\mathrm{GL}(V)$-module. (However, it turns out that it *is* indeed onto in this case.)

As this discussion suggests, sometimes representation theory helps, but it's not a panacea.