To address the limitations of existing hypergraph modeling and analysis, a new framework for analyzing the synchronization stability of higher-order networks is proposed. First, we define a two-dimensional adjacency matrix that can fully describe the topology of a hypergraph, which is composed of the sub-adjacency matrices of each hyperedge. Subsequently, based on this structure, we construct a new dynamical model for the hypergraph and derive its variational equation for synchronization stability through linearization analysis. Then, under the assumption that the coupling functions have consistent Jacobian behavior near the synchronization manifold, the complex variational equation is transformed into a standard master stability function (MSF) form, determined by a single effective coupling matrix and a Jacobian matrix, so as to reduce the strict restrictions on the hypergraph structure and coupling forms. This effective coupling matrix contains the topological information and coupling strengths of all hyperedges. Using the MSF, the relationship between system parameters and synchronization stability is analyzed, providing a novel theoretical tool for the design and control of complex networks with higher-order interactions. Finally, the validity of the theoretical results is verified through numerical simulations of two actual systems.