Suppose we have a manifold $M$ of dimension $n$ and its cotangent bundle $T^*M$. The Liouville form $\lambda$ on $T^*M$ is defined as $\lambda_{\omega_p} = \pi^*(\omega_p)$ where $\pi$ is the standard projection map from $T^*M$ to $M$. The problem is to prove that $d\lambda$ is a nondegenerate 2-form on $T^*M$. I was able to solve this problem by simply expressing the form with the aid of local coordinates and deriving the result by straightforward computation. However, I also have a bogus proof where I can't identify the incorrect step. Exponents are repeated wedge products.
$d\lambda_{\omega_p} = d\pi^*(\omega_p)$
$(d\lambda_{\omega_p})^n = (d\pi^*(\omega_p))^n = \pi^*(d\omega_p)^n$
(Because pullbacks commute with exterior differentiation and wedge products).
However, $\pi^*(d\omega_p)^n = 0$ because it is the image of a $2n$ form on a $n$ dimensional manifold.
Where am I going wrong? I'm guessing I'm missing some hypothesis in order to use the commutative fact but I am not sure. Thank you.