SO(3,q) to PGL(2,q)

Question

Can anyone suggest a reference to an explicit formula giving, in the standard matrix notation, a homomorphism from SO(3,q) to PGL(2,q) (classical matrix groups: orthogonal of dimension 3 and projective general linear of dimension 2 over finite field of order q. To write the reverse map, from PGL(2,q) to SO(3,q) is easy: this is just the adjoint representation of GL(2,q).