$$\begin{align*}
d(\iota_u\Omega)
&= d((-y - xz)\,dx + (x-yz)\,dy + (1-z^2)\,dz,)\\
&= -dy\wedge dx -d(xz)\wedge dx + dx\wedge dy - d(yz)\wedge dy + d(1-z^2)\wedge dz\\
&= 2\,dx\wedge dy + dx\wedge(z\,dx+x\,dz) + dy\wedge(z\,dy + y\,dz)\\
&= 2\,dx\wedge dy + x\,dx\wedge dz + y\,dy\wedge dz
\end{align*}$$