Different from existing methods that use matrix multiplications and have high computation complexity, this paper proposes an efficient generation method of n-dimensional (nD) Cat maps using Laplace expansions. New parameters are also introduced to control the spatial configurations of the nD Cat matrix. Thus, the proposed method provides an efficient way to mix dynamics of all dimensions at one time. To investigate its implementations and applications, we further introduce a fast implementation algorithm of the proposed method with time complexity O(n) and a pseudorandom number generator using the Cat map generated by the proposed method. The experimental results show that, compared with existing generation methods, the proposed method has a larger parameter space and simpler algorithm complexity, generates nD Cat matrices with a lower inner correlation, and thus yields more random and unpredictable outputs of nD Cat maps.