In this paper, we consider a Cartesian moving reference frame. Its angular velocity vector is introduced as a solution to a system of kinematic equations of basis vectors. These equations connect the position of the basis vectors with their velocity. The construction of a formula for the angular velocity vector of an orthonormal basis is described. It is shown that the angular velocity vector in the found form is a solution to the system of the equations. Using transformations of the constructed solution, four more representation forms of the angular velocity vector are derived. It is shown that all the obtained forms define the same angular velocity vector of the moving space, though they contain different elements. All of the forms are also solutions of the system of kinematic equations. Presented results can be applied both to a solid body and to any rigid system.