(*Compute cross product between vectors*)
aVec={a1,a2,a3};
bVec={b1,b2,b3};

(*Computing cross product using mma internal command*)
cVec//MatrixForm
cVec=Cross[aVec,bVec];

(*Computing cross product from definition*)
ϵ=LeviCivitaTensor[3]//Normal;
dVec=Table[Sum[ϵ[[i,j,k]]aVec[[j]]bVec[[k]],{j,1,3},{k,1,3}],{i,1,3}];

dVec==cVec