A new multiscale finite element method, the Multiscale Hybrid (MH) method, is proposed, analyzed, and tested in this work. The method is built as a close relative to the multiscale hybrid mixed (MHM) method, but with the fundamental difference that a novel definition of the Lagrange multiplier is introduced. The practical implication of this is that both the local problems to compute the basis functions, as well as the global problem, are elliptic, as opposed to the MHM method (and also other previous methods) that is needed to solve a mixed global problem, as well as constrained local problems to compute the basis functions. The error analysis of the method is based on a hybrid formulation, and a static condensation process is done at the discrete level, so the final global system only involves the Lagrange multipliers. The performance of the method is tested by numerical experiments for problems with multiscale coefficients, and comparisons with the MHM method are carried out in terms of performance, accuracy, and memory requirements.