It is a well-known fact that the group of orientation-preserving isometries of the hyperbolic n-space is isomorphic to the matrix group SO^+(n,1). When n=2 and n=3, these groups have a "friendlier" description as the 2x2 matrix groups PSL(2,R) and PSL(2,C). By identifying R^4 with the quaternion algebra H, we will see that something similar happens in the n=5 case: more precisely, we will show that SO^+(5,1) is isomorphic to PSL(2,H) - the space of 2x2 quaternionic matrices with Dieudonné determinant equal to 1. At the end of the talk, I will give an idea on how these results can be applied to try and understand deformations of complete hyperbolic 3-manifolds (with finite volume) in the 5-dimensional hyperbolic space. This is based on a joint work with Bruno Martelli.