R=FullSimplify[RotationMatrix[θ,{e1,e2,e3}],Assumptions->{e1∈Reals,e2∈Reals,e3∈Reals,θ∈Reals}]/.(e1^2)+(e2^2)+(e3^2)->1;
R//MatrixForm

FullSimplify[Det[R]]/.{(e1^2)+(e2^2)+(e3^2)->1}/. ((Cos[θ])^2)+((Sin[θ])^2)->1