Angular velocity and rotation matrices

I’m running into some confusion understanding rotations as being instantaneous cross products of angular velocity vectors. I would like to highlight two approaches which seem like they contradict each other, in the hopes that somebody can explain the conceptual error being made.

Approach 1

Don’t think about frames. Just imagine I have some vector rotating according to some rotation matrix $A(omega t).$
$$vec{r}(t) = A vec{r_0}$$
$$ A^T vec{r}(t) = vec{r_0}$$
Now, I want to find how this changes with time.
$$frac{d vec{r}(t)}{dt} = frac{dA}{dt} vec{r_0}$$
$$dot{vec{r}(t)} = dot{A} vec{r_0}$$
$$dot{vec{r}(t)} = dot{A} A^T vec{r}(t)$$
Now, it so happens that $dot{A} A^T $ is an antisymmetric matrix. So I can associate it with the cross product of some $vec{omega.}$
$$dot{vec{r}(t)} =vec{ omega} times vec{r}(t).$$
Approach 2

Now think about frames.
We have the same rotation matrix which transforms a basis set $vec{e}_j’=A_{ij} vec{e}_i.$

Then for any vector $vec{r}$ you have

$$vec{r}=r_j’ vec{e}_j’ = A_{ij} r_j’ vec{e}_i.$$

Now you look at how this changes with time

$$dot{vec{r}}=(A_{ij} dot{r}_j’ + dot{A}_{ij} r_j’) vec{e}_i.$$

We put this in the components of the body-fixed frame

$$dot{vec{r}} = (A_{ij} dot{r}_j’ + dot{A}_{ij} r_j’) A_{ik} vec{e}_k’ = (dot{r}_k’ + Omega_{kj} r_j’) vec{e}_k’.$$
In the case of a vector that is just rotating, this yields
$$dot{r_i}vec{e}_i= dot{A}_{ij} A_{ik} r_j’ vec{e}_k’ $$
But when we transform this back to the "space" system we get
$$dot{vec{r}(t)} = A^T dot{A} vec{r}(t).$$
So why do I get that $vec{omega} = dot{A} A^T$ in the first approach, but $vec{omega} = A^T dot{A} $ in the second approach?