A conformal frame for a conformal structure $\sigma$ of signature $p,q$ on a manifold $M$ of dimension $n=p+q$ is a pair $(m,u)$ of point $m\in M$ and linear isomorphism $T_m M\xrightarrow{u}\mathbb{R}^{p,q}$ identifying conformal structures.
The conformal frame bundle $F_{\sigma}$ is the set of all conformal frames, equipped with the map $(m,u)\mapsto m$ and the action $(m,u)h=(m,h^{-1}u)$, for $h\in CO_{p,q}$ a conformal linear transformation of $\mathbb{R}^{p,q}$.
Clearly $CO_{p,q}\to F_{\sigma}\to M$ is a smooth principal right $CO_{p,q}$-bundle.
Similarly, for each volume density $|\omega|$ on $M$ (locally represented by a volume form), a volume frame is a pair $(m,u)$ of point $m\in M$ and linear isomorphism $T_m M\xrightarrow{u}\mathbb{R}^n$ identifying the volume of open sets.
The volume frame bundle $F_{\sigma}$ is the set of all volume frames, equipped with the map $(m,u)\mapsto m$ and the action $(m,u)h=(m,h^{-1}u)$, for $h\in EL_n$ (the set of volume preserving linear isomorphisms).
Clearly $EL_n\to F_{|\omega|}\to M$ is a smooth principal right $EL_n$-bundle.
The intersection $F_{\sigma}\cap F_{|\omega|}$ is the orthonormal frame bundle of a unique pseudo-Riemannian metric of signature $p,q$.

All of this seems obvious, but it doesn't rely on choice of a representative metric.