Consider the differential 1-form $\omega$ on $S^1$ given by $$\omega_{a}(\lambda (-a_2,a_1))=\lambda$$ for all $a=(a_1,a_2)$ in $S^1$. Show that $\omega$ is closed, but not exact.
I know I have to show that $d\omega=0$ and that there is a $0$-form $\alpha$ (i.e. a smooth function) such that $d\alpha=\omega$. However, I am already struggling with the first part, since I only know how the form looks in a specific point. I thought I could write $\lambda (-a_2,a_1)=\lambda \begin{pmatrix} 0&-1\\1&0\end{pmatrix} \begin{pmatrix} a_1\\a_2 \end{pmatrix}$ and use the definition of the pullback, but that doesn't seem to help. Any help would be greatly appreciated!
Hints
(Closedness) $d\omega$ is a $2$-form on a $1$-manifold.
(Inexactness) If $\omega$ were exact, Stokes' Theorem (in this case, essentially the Fundamental Theorem for Path Integrals) would give $\int_{S^1} \omega = 0$.