... v 1 , ··· ,vk∈ Vv 1 =i 1 ϕi 1 (v 1 )ei 1 , ··· ,vk=ikϕik(vk)eik,ω(v 1 , ··· ,vk)=ω(i 1 ϕi 1 (v 1 )ei 1 , ··· ,ikϕik(vk)eik)=i 1 ,··· ,ikϕi 1 (v 1 ) ... ϕ2)(v 1 ,v2)=ϕ 1 (v 1 )ϕ2(v2) − ϕ2(v 1 )ϕ 1 (v2) = detϕ 1 (v 1 ) ϕ 1 (v2)ϕ2(v 1 ) ϕ2(v2)R2ϕ(v 1 ),ϕ(v2) ϕ =(ϕ 1 ,ϕ2):V → R2ϕ 1 , ··· ,ϕk∈ V∗kϕ 1 ∧···∧ϕk∈ Λk(V )ϕ 1 ∧···∧ϕk(v 1 , ... ,eik)ϕi 1 ∧···∧ϕik(v 1 , ··· ,vk){ϕi 1 ∧···∧ϕik, 1 ≤ i 1 < ···<ik≤ n}ϕi 1 ∧···∧ϕik(ej 1 , ··· ,ejk)= 1 (i 1 , ··· ,ik)=(j 1 , ··· ,jk)0 (i 1 , ··· ,ik) =(j 1 , ··· ,jk)ω...