This is Problem 23.5 from Tu's An Introduction to Manifolds.
Prove that the area form $\omega$ on $S^2$ in Example 23.11 is equal to the orientation form $$x dy \wedge dz - y dx \wedge dz + z dx \wedge dy$$ of $S^2$.
The hint says to take the exterior derivative of $x^2 + y^2 + z^2 = 1$ to obtain a relation among the $1$-forms $dx, dy, dz$ on $S^2$. Then show for example that for $x \neq 0$, one has $dx \wedge dy = (z/x)dy \wedge dz$. However, I do not see how to solve this problem using this hint. I would appreciate any help.