Let $M\subset (\Bbb R^3,g_0)$ be a surface (for example 2-sphere). Consider the following 1-form on $(\Bbb R^3,g_0)$: $$\omega:=ydx-xdy+zdz,$$ Then $\star\omega$ is a 2-form on $(\Bbb R^3,g_0)$. Now restrict this 1-form to $\Bbb S^2$. Hence $\star\omega|_{\Bbb S^2}$ is again a 2-form. But it must be a 1-form on sphere. I know that this can be resolve by pullback metric. But is the pullback metric is nothing except restriction of main metric to submanifold?
Question: How can I calculate $\star\omega$ on $\Bbb S^2$ directly from Euclidean metric?
My attemp: Using $xdx+ydy+zdz=0$ on 2-sphere we have $$\star(\omega\vert_{S^2})=\star((y-xz)dx+(zy-x)dy)=(y-xz)dy+(zy-x)dx,$$ Is this correct?