There's nothing special about $\mathbb C$ here except that it is a characteristic-$0$ field, so I will work over an arbitrary such field.
The identity component of a group $G$ with Lie algebra $\mathfrak{so}_n$ is $\operatorname{SO}_n$, so you're asking about the finite extensions of $\operatorname{SO}_n$ in $\operatorname{GL}_n$.
If $n = 1$, these are the finite subgroups of $\operatorname{GL}_1$, which are as you want.
If $n = 2$, these are the finite extensions of the (algebraic) torus $\operatorname{SO}_2$. Note that $\operatorname{SO}_2$ is a maximal torus in $\operatorname{GL}_2$, hence self-centralising. There is thus an injection from $\pi_0(G)$ to the Weyl group $\operatorname N_{\operatorname{GL}_2}(\operatorname{SO}_2)/{\operatorname{SO}_2}$, which is cyclic of order $2$. It is easy to check that the representatives of the non-identity coset of the Weyl group lie in $\operatorname O_2$, so your conjecture works in this case.
There is an injection from $\pi_0(G)/{\operatorname C_G(\operatorname{SO}_n)}$ to $\operatorname{Out}(\operatorname{SO}_n)$, which, since $n \ne 8$, is cyclic of order $1$ (for $n > 2$ even) or $2$ (for $n > 2$ odd).
We have that $\operatorname C_G(\operatorname{SO}_n)$ is contained in $\operatorname C_{\operatorname{GL}_n}(\operatorname{SO}_n)$. For $n > 2$, the latter equals $\operatorname Z(\operatorname{GL}_n)$ (as can be tested by computing with diagonal matrices, for example). Thus, for $n > 2$ odd, we are done.
For $n > 2$ even (with $n \ne 8$), the inner-automorphism map is an isomorphism $\operatorname O_n/{\operatorname{SO}_n} \to \operatorname{Out}(\operatorname{SO}_n)$. We thus have that $G$ is contained in $\operatorname Z(\operatorname{GL}_n)\cdot\operatorname O_n$, as desired.