Matrix-valued function: understanding 
$A(x)=(a_{i,j}(x))_{i,j}: \mathbb{R}^n\to \mathbb{R}^{m\times k}$ is a matrix-valued function.

From my understanding it is a function that maps a vector to a matrix. How does this happen? Is the 'function' a matrix of which the elements are functions? If so, are these just $\mathbb{R}^n-\mathbb{R}$-functions? 
I don't really understand the form of a matrix-valued function and how it 'takes' the required input. Maybe an example could be helpful?
Thanks a lot.

A more geometric example: let $T$ be a function that maps a vector $v$ to a rotation around the vector $v$ by $\frac{\pi}{6}$. Then you can represent this function as a function from $\mathbb{R}^n$ to $\mathbb{R}^{n \times n}$.
