If we have a four dimensional real spacetime $(M,g)$, with $g$ being a $(-+++)$ signature Lorentz-metric, and $\{\theta^0,\theta^1,\theta^2,\theta^3\}$ is a local orthornormal coframe defined in some open region $U$, it is possible to write the Einstein-Hilbert action (without the Gibbons-Hawking-York term) $$\int_M\star \mathrm{Scal} $$ locally as $$ \int_M[\mathrm{Tr}(\omega\wedge\omega\wedge\epsilon)-\mathrm{d}\ \mathrm{Tr}(\omega\wedge\epsilon)] ,$$ where $\omega$ is the $\mathfrak{o}(1,3)$-valued connection 1-form, and $\epsilon$ is the $\mathfrak{o}(1,3)$-valued 2-form constructed out of the Riemannian volume-form by writing it out in components, and contracting the last two indices with a 2-form basis (eg. $\epsilon$ is the matrix constructed out of the 2-forms $\epsilon^i_{\ j}$-s, where $\epsilon_{ij}=\sum_{k<l}\epsilon_{ijkl}\theta^k\wedge\theta^l$, and indices are raised/lowered by the Minkowski metric).
The domain of integration is understood to be symbolic (as if spacetime is not parallelizable, these quantities are not defined globally), and the configuration variable is taken to be the $\{\theta^i\}$ coframes (understood as an $\mathbb{R}^4$-valued 1-form), as these carry the same information as the metric tensor does.
This formalism is clearly local and while strictly speaking, coordinate-invariant, it is not gauge-invariant. All quantities defined here however (except maybe $\epsilon$) have global and invariant meaning if they are defined on the orthonormal frame bundle $\mathrm{OFr}(M)$, and it seems more "clean" to me to work on the principal bundle instead of juggling local frames on $M$ (obviously not for explicit calculations, just general formulations), on the other hand, aside from a few (for me) rather hard to understand papers, I have not seen anyone do this in gravity, and I am curious how to understand the EH-action on the frame bundle, if it is possible at all.
Question 1: It is clear the coframe lifts to become the solder form, while the connection forms lifts into, well, the connection form. What is $\epsilon$ on the frame bundle?
Question 2: How is it possible to lift the integral into the frame bundle, if it is possible at all? The integrand is a 4-form, which is means it would have to be integrated over a 4-dimensional submanifold of $\mathrm{OFr}(M)$, what submanifold is that? Is there some other mechanism that comes into play here?
Question 3: Well, basically, if this path I am thinking about is not walkable, then what is (if there is one) the action functional whose variation with respect to some (which?) configuration variable will reproduce the vacuum Einstein-equations, with everything being written in terms of globally defined, invariant fields (differential forms) on $\mathrm{OFr}(M)$?