As picture below ,I want to compute the (2) , because there is a singularity in $\{0\}$ and $\omega$ is closed . So ,I have $$ \int_M\omega=\int _{\partial B_1(0)} \omega $$ I think there is a singularity in the $\{0\}$ , so , I think I can't use Stock theory by $$ \int _{\partial B_1(0)} \omega =\int_{B_1(0)} d\omega $$ So, I can't compute it. How should to do it ?
On $\partial B_1(0)$, $\|x\|=1$, thus
$$\int_{\partial B_1(0)} \omega = \int_{\partial B_1(0)}\omega_0, $$
where
$$\omega_0 = \sum_{i=1}^n (-1)^{i+1} x_i dx^1 \wedge \cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^n.$$
Now the singularity is gone and you can apply Stokes' theorem.