There are four division algebras namely, real numbers, complex numbers, quaternions and octonions. They can be used to represent a number of orthogonal groups. In particular, the groups SO(3) and SO(4) of rotations of 3- and 4-dimensional spaces, respectively, can be described in terms of quaternions. We start with reviewing these cases and next turn to the groups of rotations of 7- and 8- dimensional spaces and describe them in terms of octonions. Since octonions form a non-associative division algebra, we use Moufang Identities to overcome the difficulty of some calculations and provide transformations that generate groups SO(7) and SO(8), which is an alternative description for these orthogonal groups.