Say we start with $\mathbb{R}^d \times \mathbb{R}^d$, which we interpret as a fiber bundle. Then we imagine having an orthonormal basis for each fiber. Finally, we prescribe a connection; that is, an infinitesimal rotation from $SO(d)$ -- or $SO(d-1,1)$ if this is to be pseudo-Riemannian -- between each infinitesimally close pair of points in the space. Not sure the more precise way to say that. We then claim that this space is really a Riemannian manifold, the fiber bundle is the tangent bundle, the bases constitute a frame field, and the rotation is the Levi-Civita connection, that is, the covariant derivative of the frame field.
Is this valid? Can we always determine a Riemannian metric that has the given connection, and is it unique? Or does it work as long as certain topological requirements are fulfilled? Or does it not work in general?
EDIT: Thanks to Didier's comment that you can scale the metric without changing the connection. So then, is it unique up to scaling?
