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This paper presents a discrete-time rigid body motion model using Cayley map for the special Education group SE(3). A continuous-time model and related discretization methods are introduced to illustrate the motivation of this work. The Cayley map for SE(3) is then adopted as a vector representation of an element of SE(3). The proposed representation allows exact computation of the discrete-time equation of rigid body kinematics. Moreover, the gradient of functions with respect to the state or input can also be easily computed. An application to an optimal control of a fully-actuated system on SE(3) is considered, and the gradient of the cost function is computed analytically. A simulation shows that the preset method dramatically improves the computation time.