How do we construct a precise map of inner + outer automorphism of special orthogonal group $SO(n;\mathbb{R})$?

- $d=2$; We can look at $SO(2;\mathbb{R})=U(1)$ which is abelian, and we know the inner
$$
Inn(SO(2;\mathbb{R}))=SO(2;\mathbb{R})/Z(SO(2;\mathbb{R}))=1
$$
$$
Out(SO(2;\mathbb{R}))=\mathbb{Z}/2
$$
The total $Aut(SO(2;\mathbb{R}))=Inn(SO(2;\mathbb{R})) \rtimes Out(SO(2;\mathbb{R}))=\mathbb{Z}/2$
We have no $Inn(SO(2;\mathbb{R}))$ except the identity map.
I believe that we can get the $Out(SO(2;\mathbb{R}))=\mathbb{Z}/2$ by flipping $t \to -t$ in
$$
U(1)=\{\exp(i t) | t \in [0, 2 \pi)\}
\to
\{\exp(-i t) | t \in [0, 2 \pi)\}.
$$
I wish to see explicit answer like the above for my following questions ---

- other $n$ but $n\neq 2,8$ is discussed in https://math.stackexchange.com/q/3843014/141334 with answer still pending.

- for $n=8$  


> Question 1: How do we construct the inner automorphism map explicitly  (if my result is correct?)

for $n=8$

$$
Inn(SO(n;\mathbb{R}))=SO(n;\mathbb{R})/Z(SO(n;\mathbb{R})) = 
SO(8;\mathbb{R})/\mathbb{Z}/2 
$$

> Question 2: How do we construct the outer automorphism map explicitly 
$$
Out(SO(8;\mathbb{R}))=S_3
$$
Given the parametrization of  $SO(n;\mathbb{R})$ how to map to itself via the Out map?


> Question 3: How do we construct the total automorphism map explicitly 
$$
Aut(SO(8;\mathbb{R}))=Inn(SO(8;\mathbb{R})) \rtimes Out(SO(8;\mathbb{R}))
=(SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes  S_3?
$$
Given the parametrization of  $SO(8;\mathbb{R})$ how to map to itself via the Aut map?


p.s. Possible useful link but with not explicit (not enough) constructions in https://mathoverflow.net/q/235758/106497