Given a geometric surface (as defined in O'Neill in my understanding simply a Riemannian surface) $M$, the connection form is simply defined by the $\omega_{12}$ that satisfies the following equations:
$$ d\theta_1 = \omega_{12} \land \theta_2 \\ d\theta_2 = \omega_{21} \land \theta_1 $$
But how do I actually compute it? For example, in the following problem:
Problem 7. Let $\mathbb{H}=\{(u, v): v>0\}$ be the hyperbolic plane and use $\mathbf{x}(u, v)=(u, v)$ as the trivial coordinates. Consider the hyperbolic metric $$ \left\langle\mathbf{x}_{u}, \mathbf{x}_{u}\right\rangle=\frac{1}{v^{2}}, \quad\left\langle\mathbf{x}_{u}, \mathbf{x}_{v}\right\rangle= 0, \quad\left\langle\mathbf{x}_{v}, \mathbf{x}_{v}\right\rangle=\frac{1}{v^{2}} $$
I am at a great loss to see how to actually derive $\omega_{12}$ for this surface (let alone in general). Any hints?