In four-dimensional or higher-dimensional space, spin angular momentum is not known well. For instance, we are not sure how many components it has in four-dimensional space. In this paper, the spin properties are theoretically derived and investigated in general-dimensional space. Before studying spin, we review the orbital angular momentum properties first, because they are theoretically well known even in higher-dimensional space. Then, for the orbital angular momentum, we make a special but natural assumption, and see that it leads a quantity similar to spin or Pauli matrices. It is remarkable that all the spin properties are derived from the one and only one assumption. Consistency of our consideration is checked by calculating the four-dimensional version of spin spherical harmonics and its eigenvalues. This theory may enable us to understand the double degeneracy of spin-1/2 particles without bringing in something called spin.