Can this group be written as a semidirect product?

Question

Assume, $\mathbb{F}$ is a field of $char=0$ and $\mathbb{F}^\ast$ be the group of non-zero scalars. Let $G=\{(c,A)\in \mathbb{F}^\ast \times GL_n(\mathbb{F}) | AA^t=cI_{n\times n} \}$. Then $G$ is a group with the obvious entry wise multiplicaton. Can we write $G$ as a suitable semidirect product? If not in general, can we do this for the complex numbers in particular? For the real numbers $\mathbb{R}$ it seems $G$ can be written as $\mathbb{R}_{> 0} \ltimes O_n$, as follows from the following split exact sequence: $1\to O_n \xrightarrow{i} G \xrightarrow{p} \mathbb{R}_{> 0} \to 1 $, here $i$ is $A \mapsto (1, A)$ and $p$ is $(c, A) \mapsto c$. A splitting map $s: \mathbb{R}_{> 0}\to G$ can be given by $s(c):= (c, \sqrt cI)$. So the question is whether we can write the above defined $G$ as a suitable semidirect product in more general situations, and if not why?