Write the coordinates on $ \mathbb {R} ^{2n+1}$ as $ \displaystyle{ (x_1 , y_1, x_2, y_2, \cdots ,x_n, y_n ,z)}$. Define the 1-form $ \displaystyle{ \omega:= dz +x_1 \, dy_1+ x_2 \, dy_2 + \cdots + x_n \, dy_n} $.
Compute $ \displaystyle{ \omega \wedge (d \omega \wedge d \omega \wedge \cdots \wedge d \omega )}$ where the wedge product is taken n times.
I first work out the simply cases $n=1,2,3$ and I guess that it must be
$ \displaystyle{ \omega \wedge (d \omega \wedge d \omega \wedge \cdots \wedge d \omega ) =n dz \wedge dx_1 \wedge dy_1 \wedge dx_2 \wedge dy_2 \wedge \cdots \wedge dx_n \wedge dy_n }$
but I have no proof for the general case.