I would like to find a 2-form on $\mathbb{R}^3$ that 1) restricts to the torus to give a generator of the top-dimensional de Rham cohomology of the torus and 2) restricts to the torus to give the surface area of the torus when integrated over it, for the sake of specificity, a torus with larger radius 3 in the $x-y$-plane and smaller radius 1, $T^2(3,1)$.
Per this webpage, https://mathworld.wolfram.com/Torus.html, a parameterization for $T^2(3,1)$ is given by
$\phi :\begin{cases} x = [3+\cos(v)]\cos(u) \\ y = [3+\cos(v)]\sin(u) \\ z = \sin(v) \end{cases}$,
$u \in [0,2\pi], v \in [0,2\pi]$
Per my musings on this webpage, http://airvigilante194.sdf.org/mathjax/orientedBordism.html, I came up with 1-forms, [EDIT: I think there should be a factor of $\displaystyle \frac{1}{3}$ on each variable in $\omega_1$, so it is the generator of $H^1_{dR}(S^1)$ on the circle of radius 3 which returns the "length" when integrated over that circle]
$\displaystyle \omega_1 = \frac{-\frac{1}{3}y}{\left(\frac{1}{3}x\right)^2+\left(\frac{1}{3}y\right)^2}\ dx + \frac{\frac{1}{3}x}{\left(\frac{1}{3}x\right)^2+\left(\frac{1}{3}y\right)^2}\ dy = \frac{-3y}{x^2+y^2}\ dx + \frac{3x}{x^2+y^2}\ dy$ and
$\displaystyle \omega_2 = \left[\frac{-z}{z^2 + \left(\frac{y^2}{x^2+y^2}\right)(y-3)^2 + \left(\frac{x^2}{x^2+y^2}\right)(x-3)^2}\right]\ \left[\left(\frac{y}{\sqrt{x^2+y^2}}\right)dy + \left(\frac{x}{\sqrt{x^2+y^2}}\right)dx\right] + \left[\frac{\left(\frac{y}{\sqrt{x^2+y^2}}\right)(y-3)+ \left(\frac{x}{\sqrt{x^2+y^2}}\right)(x-3)}{z^2 + \left(\frac{y^2}{x^2+y^2}\right)(y-3)^2 + \left(\frac{x^2}{x^2+y^2}\right)(x-3)^2}\right]\ dz$,
(taking linear combinations of the $(x-3)$- and $(y-3)$-variables in $\omega_2$)
and a 2-form $\eta = \omega_1 \wedge \omega_2$.
I think $\eta$ should be a 2-form on $\mathbb{R}^3$ that restricts to give a 2-form on $T^2(3,1)$ whose integral over $T^2(3,1)$ is its surface area, but I'm very unsure about $\omega_2$.
EDIT: Again per https://mathworld.wolfram.com/Torus.html, in the parameterization $\phi$ of the torus, this should be the pull-back of the plain-vanilla surface area form on the torus $\theta = \iota^*(dx \wedge dy \wedge dz)$ (the pull-back of the plain-vanilla volume form $dx \wedge dy \wedge dz$ on $\mathbb{R}^3$ via the inclusion map $\iota: T^2(3,1) \hookrightarrow \mathbb{R}^3$) to the parameter space
$\phi^*(\theta) = [3+\cos(v)]\ (du \wedge dv)$
and if I integrate the constant function 1 over the torus in the parameterization $\phi$, I should get for the surface area
$\displaystyle \iint\limits_{T^2(3,1)} 1\ dS = \int_0^{2\pi}\int_0^{2\pi} [3+\cos(v)]\ du\ dv = 12\pi^2$
I made up a wxMaxima script to try to compute the integral of $\eta$ via a pull-back along $\phi$, http://airvigilante194.sdf.org/mathjax/Form%20Integrals%2001.wxmx, but it keeps crashing, likely because I'm doing something very wrong with $\omega_2$.
I know I should be able to puzzle this out myself, but could someone help me come up with the correct explicit formula for $\omega_2$? I would be very grateful.