Computing $\alpha\wedge d\alpha$ where $\alpha = dz + x\,dy$

Question

Let $\alpha = dz + xdy$ be a one-form on $\mathbb R^3$. I would like to compute the wedge product $\alpha\wedge d\alpha$ as explicitly as possible but I am not sure if I am doing this correctly.

$$d\alpha = d(dz+x\,dy) = dx \wedge dy$$

so

\begin{align*} \alpha\wedge d\alpha &= (dz+x\,dy)\wedge(dx\wedge dy)\\ &= dz\wedge dx\wedge dy + x\,dy\wedge dx\wedge dy\\ &= dz\wedge dx\wedge dy\\ &= dx\wedge dy\wedge dz. \end{align*}

Is this correct ?

Thank in advance.