This is usually called the
cocycle description of vector bundles; this data is precisely what is called a Cech cocycle in $Z^1(M; \mathcal{GL}_n)$, where the latter is the sheaf of groups given by $\mathcal{GL}_n(U) = \text{Map}(U, GL_n)$; one can perfectly make sense of first cohomology with coefficients in sheaves of nonabelian groups, and what the above description says is $$H^1(M; \mathcal{GL}_n) \cong \text{Vect}_n(M).$$