As suggested by [Denis Nardin](http://mathoverflow.net/users/43054/denis-nardin) I am moving my comment here. However I don't know details well enough, so I am making this cw in case somebody can fill them in.

So, choose a cohomology theory $h^*$ like e. g. $K$-theory, and, given a cover $(U_i)_{i\in I}$ of $X$, let $p:Y\to X$ be the canonical map $\coprod_{i\in I}U_i\to X$. Then consider the associated simplicial space
$$
\check C(p):=\left(Y\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix} Y\times_XY\begin{smallmatrix}\leftarrow\\\rightarrow\\\leftarrow\\\rightarrow\\\leftarrow\end{smallmatrix}Y\times_XY\times_XY\begin{smallmatrix}\leftarrow\\\vdots\\\leftarrow\end{smallmatrix}\cdots\right),
$$
so that $\check C(p)_n=\coprod_{i_1,...,i_n}U_{i_1}\times_X\cdots\times_XU_{i_n}$.

Under certain (I believe mild) conditions the induced map from its geometric realization to $X$ induces isomorphism on $h^*$. On the other hand - see e. g. Theorem 5.83 on [page 163](https://books.google.com/books?id=3HY4ruJ6BigC&pg=PA163&redir_esc=y#v=onepage&q&f=false) of "Generalized Cohomology" by  Kōno and Tamaki - there is a spectral sequence converging (again under some mild conditions) to $h^*$ of the geometric realization, with the second page given by cohomology of the cochain complex corresponding to the cosimplicial abelian group $h^*(\check C(p))$.

In our case - yet again under some conditions - the Moore normalization of the latter complex (given by restricting to nondegenerate simplices of $\check C(p)$) has $\prod_{i_1,...,i_n}h^*(U_{i_1}\cap\cdots\cap U_{i_n})$ in the $n$th degree, with understandable differentials.