How to find the axis of rotation or location given the angular velocity?

Your formula itself is wrong.

$$ frac{dvec{r}}{dt} = vec{v} = omega times vec{r}$$

Angular Velocity ($omega$) is an axial vector whereas velocity is a polar vector.

Axial vectors are vectors whose directions oriented in a non-intuitive way. For example, when you rotate a disc by a certain angle ($dphi$), the $dphi$ vector lies along the axis of rotation.

A polar vector is a vector whose direction makes sense to a layman. For example, the direction of the velocity vector is in the direction of motion of the body.

Lets begin with rotation in a plane. You have the intuition for polar vectors, so if you know the direction of the velocity vector and the radius vector, you can use the right-hand rule to find the direction of the angular velocity vector.

$$vec{v} = omega times vec{r}$$

If you do it correctly, you will find the angular velocity vector to be oriented along the axis of rotation.

For cases where you have the rotation in more than one dimension. You can split the angular velocity vector into two mutually perpendicular components. These components are not along the axis of rotation.

You need two things. The angular velocity vector $vec{omega}$ at some time frame, and the linear velocity vector at some reference point $vec{v}$.

The axis of rotation is located at a point relative to the reference point

$$ vec{r} = frac{vec{omega} times vec{v}}{ | vec{omega} |^2} tag{1}$$

You can apply this at every instance.

See related post1 post2 and post3.

_{The proof is easy but it requires the use of the vector triple product identity $a times (b times c) = b (ac) - c (ab)$. Substitude above $vec{v} = vec{r} times vec{omega}$ and carry out simplifications.}

Simple example

The center of mass of an object is moving with $vec{v} = pmatrix{5 & -1 &0}$ and the rotational vector is $vec{omega} = pmatrix{0 & 0 & 2}$ at some instant.

The point on the axis of rotation closest to the center of mass is located at $$vec{r} = frac{pmatrix{0&0&2} times pmatrix{5&-1&0}}{ | pmatrix{0&0&2} |^2 } = frac{ pmatrix{2 & 10 & 1} }{2^2} = pmatrix{0.5 & 2.5 & 5}$$

To verify find the velocity at the center of mass with

$$ vec{v} = vec{omega} times (-vec{r} ) = vec{r} times vec{omega} = pmatrix{0.5 & 2.5 & 0} times pmatrix{0&0&2} = pmatrix{5&-1&0} ;checkmark $$

NOTE: That the axis of rotation is any point along the line, which includes $vec{r}$ and all points parallel to $vec{omega}$.