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The full spatial covariance matrix of the multiple-input multiple-output channel is an important quantity that is used in channel modeling, communication system signal processing, and performance analysis. This paper formulates a closed-form expression for this covariance matrix based on the average multipath propagation characteristics and the radiation patterns of the antennas in the transmit and receive arrays. Simplifications to the formulation are established for cases where the propagation can be represented using discrete plane waves or where the power angular spectrum can be modeled using clusters that satisfy uniform, truncated Gaussian, or truncated Laplacian distributions. Numerical studies demonstrate application of the formulation as well as its accuracy under different scenarios.