(OverHat[e])={0,0,1};

R=RotationMatrix[θ[t],(OverHat[e])];
R//MatrixForm

Rdot=D[R,t];
Rdot//MatrixForm

Ω=FullSimplify[Rdot.R\[Transpose]];
Ω//MatrixForm

ϵ=LeviCivitaTensor[3];

(*ω=-(1/2)(Subscript[ϵ, ijk])(Subscript[Ω, jk]),the dummy indices go into sum and the free indices go to Table*)

ω=Table[Sum[-(1/2)ϵ[[i,j,k]]Ω[[j,k]],{j,1,3},{k,1,3}],{i,1,3}];
ω//MatrixForm