v[0]={vi1,vi2,vi3};
v[1]={vj1,vj2,vj3};
v[2]={vk1,vk2,vk3};
v[3]={vl1,vl2,vl3};
v[i_,j_,k_,l_]:=Det[{v[j]-v[i],v[k]-v[i],v[l]-v[i]}];
D[v[0,1,2,3],{v[0]}]==(v[3]-v[1])\[Cross](v[2]-v[1])//Simplify
D[v[0,1,2,3],{v[1]}]==(v[2]-v[0])\[Cross](v[3]-v[0])//Simplify
D[v[0,1,2,3],{v[2]}]==(v[3]-v[0])\[Cross](v[1]-v[0])//Simplify
D[v[0,1,2,3],{v[3]}]==(v[1]-v[0])\[Cross](v[2]-v[0])//Simplify