I know that you can extract the equation of a plane in n-dimension using the two vectors, since the plane will be a linear combination of those two vectors. But I am wondering how would I get a vector other than those involved in plane formation to rotate on that plane with a given angle of rotation.
I am stuck on this thought, and have tried few things, but they are not satisfactory. Any help/insight is much appreciated.
One way to handle this easily is writing the rotation using an orthonormal basis of your plane. That is, suppose your plane is generated by the linearly independent vectors $\mathbf{v}$ and $\mathbf{w}$ of the Euclidean space $\mathbb{R}^n$, then first transform this basis to an orthonormal one, by example setting
$$ \mathrm{\mathbf e}_{1}:=\frac{\mathbf{v}}{\|\mathbf{v}\|},\quad \mathrm{\mathbf e}_{2}:=\frac{\mathbf{w}-\langle \mathbf{w},\mathrm{\mathbf e}_{1} \rangle \mathrm{\mathbf e}_{1} }{\|\mathbf{w}-\langle \mathbf{w},\mathrm{\mathbf e}_{1} \rangle \mathrm{\mathbf e}_{1}\|}\tag1 $$
where $\langle \,\cdot\, ,\,\cdot\, \rangle$ is the dot product and $\|\mathbf{r}\|:=\sqrt{\langle \mathbf{r},\mathbf{r} \rangle }$ is the induced norm. Then any vector on the plane generated by $\mathbf{v}$ and $\mathbf{w}$ can be written as $\lambda \mathrm{\mathbf e}_{1}+\mu \mathrm{\mathbf e}_{2}$ for scalars $\lambda ,\mu\in \mathbb{R}$, so using the ordered basis $B:=\{\mathrm{\mathbf e}_{1},\mathrm{\mathbf e}_{2}\}$ you can write these vectors as $(\lambda ,\mu)_B$.
Now to rotate this vector you only need to rotate it in the same way as you will do in $\mathbb{R}^2$, by example to rotate an angle $\alpha $ counterclockwise (respect to the orientation given by $B$) you do
$$ \begin{bmatrix} \cos \alpha & -\sin \alpha \\\sin \alpha &\cos \alpha \end{bmatrix}\begin{bmatrix} \lambda \\\mu \end{bmatrix}\tag2 $$
Now the result of (2) will be the rotated vector written in basis $B$.