I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with the first and second fundamental forms in a classical differential geometry course where I studied regular curves and surfaces in $\mathbb{R}^3$, but am trying to adapt this knowledge to the study of smooth manifolds using the theory of differential forms.
Recall that the structure equations for a given frame field $(\mathbf{x},e)$ are
$$d\omega^i = -\sum_{j=1}^3 \omega_j^i \wedge \omega^j, \qquad d\omega_j^i = -\sum_{k=1}^3 \omega_k^i \wedge \omega_i^k,$$
where $\omega^i = d\mathbf{x} \cdot \mathbf{e}_i$ and $\omega_i^j = d\mathbf{e}_i \cdot \mathbf{e}_j$.
Now, let $M$ be a surface and let
$$\mathbf{x} : \begin{pmatrix} x^1 \\ x^2 \\ x^3 \end{pmatrix} : M \to \mathbb{R}^3$$
be a smooth immersion. An Euclidean frame field along $\mathbf{x}$ on an open set $U \subset M$ consists of smooth vector fields $\mathbf{e}_i : U \to \mathbb{R}^3$, $i = 1,2,3$, such that $e = (\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3)$ is an orthonormal frame of $\mathbb{R}^3$ at each point of $U$. Then $d\mathbf{x}$ can be expressed in terms of $e$ by
$$d\mathbf{x} = \sum_{i=1}^3 \omega^i \mathbf{e}_i,$$
where each $\omega^i$ is a smooth $1$-form defined on $U$ given by $$\omega^i = d\mathbf{x} \cdot \mathbf{e}_i,$$ which is a linear combination of the $dx^i$ with coefficients being the smooth funtions entries of $\mathbf{e}_i$. One can check that choosing $\mathbf{e}_3$ normal at every point is equivalent to the condition that $\omega^3 = 0$ at every point of $U$. This is what we call a first-order frame field along $\mathbf{x}$.
By Cartan's Lemma, a first order frame field satisfies
$$\omega_i^3 = \sum_{j=1}^2 h_{ij} \omega^j, \qquad \text{for $i=1,2$ and $h_{12}=h_{21}$}.$$
Going back to the theory of surfaces of Gauss, we see that for this frame field, $I$ and $II$ are given by
\begin{align} I &= d\mathbf{x} \cdot d\mathbf{x} = \omega^1\omega^1 + \omega^2\omega^2, \\ II &= -d\mathbf{x} \cdot d\mathbf{e}_3 = \omega^1\omega_1^3 + \omega^2\omega_2^3 \\ &= h_{11}\omega^1\omega^1 + 2h_{12}\omega^1\omega^2 + h_{22}\omega^2\omega^2. \end{align}
Knowing this, my question is
Given a smooth function $f : \mathbb{R}^2 \to \mathbb{R}^3$ such that $M = \Gamma(f)$ defines a smooth surface immersed in $\mathbb{R}^3$, what is its second fundamental form $II$ in term of differential forms? How do I calculate the $h_{ij}$'s in this case?
I am not sure where to start. This seems like a reasonably easy exercise though. Any help would be greatly appreciated.