I had a pretty simple question but was having trouble finding the answer anywhere. If I have an orthogonal matrix $A: \mathbb{R}^n \to \mathbb{R}^n$, it should induce an automorphism on the tangent bundle as well. Is there a general formula I can find to find it. I'm imagining that if $v$ is a section, then the induced $\phi:T\mathbb{R}^n \to T\mathbb{R}^n$ should be
$$ (\phi v)(x) = Av(A^{-1}x)$$
Is this correct?
I would tend to write a point in $T\mathbb{R}^n$ as a pair $(x,v)$, where $x\in\mathbb{R}^n$ and $v\in T_x\mathbb{R}^n\cong\mathbb{R}^n$. You could then define a bundle automorphism $\phi:T\mathbb{R}^n\rightarrow T\mathbb{R}^n$ by $$\phi(x,v)=(x,Av).$$