Alternative way:

Needs["VectorAnalysis`"]

GenEx1 = { AA[Xx,Yy,Zz],BB[Xx,Yy,Zz],CC[Xx,Yy,Zz]};
GenEx2 = { DD[Xx,Yy,Zz],EE[Xx,Yy,Zz],FF[Xx,Yy,Zz]};
h0=GenEx1 . Curl[GenEx2]

From scratch

coords={Xx,Yy,Zz};
extDer[form_,i_,j_]:=D[form[[j]], coords[[i]]]-D[form[[i]], coords[[j]]]
hel[formA_, formB_]:=Module[{},
res=1/2 Sum[
formA[[ii]] extDer[formB,jj,kk] Signature[{ii,jj,kk}]
,
{ii,1,3},
{jj,1,3},
{kk,1,3}];
Simplify[res]
]

Simplify[hel[GenEx1,GenEx2]-h0]

First example of compatible forms

alpha={0,ConstC Xx,1};    (* alpha = dz + C x dy *)
beta={ConstD Yy,0,1}         (* beta = dz + D y dx *)

(*  alpha v dalpha *)
hel[alpha, alpha]

(*  alpha v dbeta *)
hel[alpha, beta]

(*   beta v dalpha *)
hel[beta, alpha]

(*  beta v dbeta *)
hel[beta, beta]

Example of contact forms that _are not_ compatible

beta={0,lambda1 Sin[K Xx],Cos[K Xx]}; (* lambda = +1/-1 *)

gamma={0,- Cos[K Xx],lambda1  Sin[K Xx]}; (* lambda = +1/-1 *)

(*  alpha v dalpha *)
hel[beta, beta]

(*  alpha v dbeta *)
hel[beta, gamma]
%/.{lambda1^2 -> 1}

(*   beta v dalpha *)
hel[gamma, beta]
%/.{lambda1^2 -> 1}

(*  beta v dbeta *)
hel[gamma, gamma]

Example of compatible contact forms that _are_ compatible

beta={0, pm Sin[kval Xx], Cos[kval Xx]};

gamma={0,  pm Sin[kval Xx + degα], Cos[kval Xx + degα]};

hel[beta, beta]

hel[beta, gamma]

hel[gamma, beta]

hel[gamma, gamma]

Simplify[ Sin[kx] Sin[kx + ϕ] + Cos[kx] Cos[kx + ϕ] ]