Near a point where $X\ne 0$, choose a coordinate system $(u_i)$ with $\partial_{u_1} = X$, then $i_Xdu_1=1$ does it. It is even exact. But then $\eta_n = du_1\wedge i_X\eta_n = du_1\wedge d\eta_{n-1}$ is exact.
Peter Michor
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