matrix Lie group embedding as a manifold

Question

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into $\mathbb{R}^n$ as a manifold, so that the manifold is generated in the same way by exponential map from the tangent space of a point, so that every element in the Lie algebra corresponds to a geodesic segment? Also if the map is not uniquely determined, it's fine anyway for my porposes. Thanks for any suggestion.

Answer 1

You get this for free when the Lie group is a matrix group, as the space of $n\times n$ matrices is homeomorphic to $\mathbb R^{n^2}$ when we consider the entries to be coordinates.