Given a connected Lie Group $G$ is there a standard method for constructing the metric tensor?
One can of course expand a group element $g=e^{x^ie_i}$ where $e_i$ are basis vectors of the Lie algebra about the identity up to linear order and compute the metric in this manner, but is there a more direct way?
An example would also be appreciated.
There is such a (bi-invariant) metric on a Lie group (isomorphic to a compact Lie group times $\mathbb{R}^{n}$). It is the one induced by the Killing form. For more input on this, see the mathoverflow discussion on the topic:
https://mathoverflow.net/questions/32554/why-the-killing-form
It should be pointed out that you do not need the exponential map to construct the metric, which is defined on the tangent space $T_{p}g, g\in G$, not on the group itself. All you need is a nicely behaved symmetric bilinear form under the group action. The explicit construction can be found at here.
Update: Yor pointed out in the comments that for a non-semisimple compact Lie group there is a bi-invariant metric, but it is not induced by the Killing form, as the Killing form degenerates on the center. To solve this, write $\mathfrak{g}=\mathfrak{a}\times \mathfrak{s}$. The Killing form induces a metric on $\mathfrak{s}$ and we put an arbitrary metric on $\mathfrak{a}$.