It is known that
$$ \exp : \mathbb{M}(n,\mathbb{C}) \to \operatorname{GL}(n,\mathbb{C}) $$
This relationship only works over the complex field.
My question is:
$$ \exp : \mathcal{G}_n(\mathbb{R})\to ? $$
where $\mathcal{G}_n(\mathbb{R})$ is the set of all real multivectors of a Clifford algebra of dimension $n$.
I know that taking the exponential of a multivector produces an invertible multivector due to $\exp V \exp -V = I$. I wonder if it is group isomorphic to $\operatorname{GL}(n,\mathbb{R})$:
$$ \exp ( \mathcal{G}_n(\mathbb{R}) )\cong \operatorname{GL}(n,\mathbb{R}) ? $$