You have several choices when choosing how to represent a rotation in 3D space. Representations mostly differ in
- number of parameters (space) required
- globality (capacity to express *every* rotation) and singularities
- readability to humans
but more minor characteristics exist like complexity, multiplicity (more parametrizations for the same physical rotation) and numerical behaviour.
The most useful rotation representations to know of for practical use are the Direction Cosine Matrix and the Quaternion. Wikipedia has a rough introduction on more of the other existing representations.