I seem to be writing down complete nonsense.
So for $(u,v)\in (0,2\pi)\times (0,2\pi)$ we take $$\Phi(u,v)=(\cos u,\sin u, \cos v,\sin v)$$ and then where it's defined $\Phi^{-1}(x,y,z,w)=(\arccos(x),\arcsin(y))$
in which case $(\Phi^{-1})^*(dx^1\wedge dx^2)=d(\arccos(x))\wedge d(\arcsin(y))=-\frac{1}{\sqrt{1-x^2}}dx \wedge \frac{1}{\sqrt{1-y^2}} dy$
where I feel like I'm messing everything up somehow lol