Let L be a geometric lattice. We first give the concept of monoidal cosheaf on L, and then define the generalized Orlik-Solomon algebra A * (L, C) over a commutative ring with unit, which is built by the classical Orlik-Solomon algebra and a monoidal cosheaf C as coefficients. Furthermore, we study the cohomology of the complement M(A) of a manifold arrangement A with geometric intersection lattice L in a smooth manifold M without boundary. Associated with A, we construct a monoidal cosheaf Ĉ(A), so that the generalized Orlik-Solomon algebra A * (L, Ĉ(A)) becomes a double complex with suitable multiplication structure and the associated total complex T ot(A * (L, Ĉ(A))) is a differential algebra, also regarded as a cochain complex. Our main result is that H * (T ot(A * (L, Ĉ(A)))) is isomorphic to H * (M(A)) as algebras. Our argument is of topological with the use of a spectral sequence induced by a geometric filtration associated with A. As an application, we calculate the cohomology of chromatic configuration spaces, which agrees with many known results in some special cases. In addition, some explicit formulas are also given in some special cases.