Demonstrate that the set of all function which takes pairs of numbers to a field, is the vectorspace of all m by n matrices

I am confused by a question in my course book.

The set of all functions $f : S → F$ from a set $S$ into a field $F$ is a vector space over $F$ relative to the following operations:

$(f + g)(s) = f(s) + g(s)$

$(af)(s) = af(s), s ∈ S$

Show that with $S = \{(i, j) | 1 ≤ i ≤ m,~ 1 ≤ j ≤ n\}$, the vector space formed is $F_{m\times n}$ (the set of all $m\times n$ matrices whose entries are scalars from $F$, with matrix addition and scalar multiplication).

I understand $S$ to be a set of all possible pairs of numbers $(1...n, 1...m)$.

How is it that the set of all functions which takes pairs of numbers to a field $F$ represents the vector space $F_{m\times n}$?

Because each function $f\colon S\longrightarrow F$ corresponds to the $m\times n$ matrix$$\begin{bmatrix}f(1,1)&f(1,2)&\ldots&f(1,n)\\f(2,1)&f(2,2)&\ldots&f(2,n)\\\vdots&\vdots&\ddots&\vdots\\f(m,1)&f(m,2)&\ldots&f(m,n)\end{bmatrix}.$$