As we know the universal covering group of $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$ is a Lie group which cannot be faithfully represented by a finite dimensional matrix.
For example, for $N>2$, the (double) covering group of the general linear group is the metalinear group $ML(N,\mathbb{R})$. The metalinear group is a subgroup of the metaplectic group $Mp(2N,\mathbb{R})$ in twice the dimension.
Therefore what's the sufficient and necessary condition in which the universal covering group of some matrix Lie group is still a matrix Lie group?