It says on Wikipedia:
"to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering."
We can turn $2 \times 2$ matricies over the reals into a Lie algebra by defining $[a, b] = ab - ba$. Which Lie group does this define? We have
$$[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0$$
for any $a, b, c$.
It is the Lie group $\mathrm{GL}(2,\mathbb R)$. The Lie algebra $\mathfrak{gl(2,\mathbb R)}$ is the space of all $2\times 2$ matrices.