# Principled manner to rotate vector in a ball in $\mathbb{R}^n$

I think this is a commonly asked question but I haven't found a satisfying answer to this.

Say we have unit non-negative vectors $$w_1,w_2 \in \mathbb{R}^2$$. I want to get all the vectors $$x$$ on the unit circle between $$w_1$$ and $$w_2$$. One way to trace the path from $$w_1,w_2$$ would be to use Givens rotation matrices to "rotate" $$x$$ from $$w_1$$ to $$w_2$$. As an example, say $$w_1$$ subtends angle $$\theta_1$$ with the first axis and $$w_2$$ subtends $$\theta_2 > \theta_1$$. Then I can rotate $$x$$ from $$w_1$$ to $$w_2$$ by using the following rotation for all angles $$\alpha \in \left[0,\theta_2-\theta_1\right]$$.

$$\begin{bmatrix} \cos{\alpha} &-\sin{\alpha} \\ \sin{\alpha} & \cos{\alpha} \end{bmatrix} \begin{bmatrix} \cos{\theta_1} \\ \sin{\theta_1} \end{bmatrix} = \begin{bmatrix} \cos{(\theta_1 + \alpha)} \\ \sin{(\theta_1 + \alpha)} \end{bmatrix}$$

Can we generalise this to $$\mathbb{R}^n$$. Say we have unit non-negative vectors $$w_1, w_2, \ldots ,w_n \in \mathbb{R}^n$$. These vectors would form a surface on the unit ball. Is there a parameterised way I can "rotate" a vector to trace out this surface. What do these rotation matrices look like?