In this work, we study a family of wireless channel simulation models called geometry-based stochastic channel models (GBSCMs). Compared to more complex ray-tracing simulation models, GBSCMs do not require an extensive characterization of the propagation environment to provide wireless channel realizations with adequate spatial and temporal statistics. The trade-off they achieve between the quality of the simulated channels and the computational complexity makes them popular in standardization bodies. Using the generic formulation of the GBSCMs, we identify a matrix structure that can be used to improve the performance of their implementations. Furthermore, this matrix structure allows us to analyze the spatial covariance of the channel realizations. We provide a way to efficiently compute the spatial covariance matrix in most implementations of GBSCMs. In accordance to wide-sense stationary and uncorrelated scattering hypotheses, this covariance is static in frequency and does not evolve with user movement.