Can someone check if my work here is on the right track:
Compute $$\int_{S^{1} \times S^{1}} f^{*} \mathbb{\omega}$$ where $f\colon S^{1} \times S^{1} \rightarrow \mathbb{R}^{4}$ is a smooth map and $\mathbb{\omega} = dx_{1} \wedge dx_{2} - dx_{3}\wedge dx_{4}$ is a two-form on $\mathbb{R}^{4}$ in the standard coordinates.
Here is what I have done so far. Let, $f(u,v) \rightarrow(x_{1}(u,v), x_{2}(u,v), x_{3}(u,v), x_{4}(u,v))$. So $$dx_{1} = \frac{\partial x_{1}}{\partial u} du + \frac{\partial x_{1}}{\partial v} dv$$ and we can do the same for $dx_2, dx_3$ and $dx_4$. So I get:
$$\int_{S^{1} \times S^{1}} \left(\frac{\partial x_{1}}{\partial u} \frac{\partial x_{1}}{\partial v} - \frac{\partial x_{2}}{\partial u} \frac{\partial x_{2}}{\partial u}\right) du \wedge dv + \left(\frac{\partial x_{4}}{\partial u} \frac{\partial x_{4}}{\partial v} - \frac{\partial x_{3}}{\partial u} \frac{\partial x_{3}}{\partial v}\right) du \wedge dv, $$
but I am not sure where to go from here. Any tips?