Is matrix just a representation? Is that precise enough?

From what i know, a $m \times n$ matrix in which elements are taken from a field like $F$, is a rectangular array which has $m$ rows and $n$ columns. That's the definition i learned from linear-algebra class.

Every time i want to work with matrices, I simply draw that rectangular array and work with it. But it seems not precise and accurate enough to me. My argue is that, "What is a rectangular array exactly?"

In computer science, a rectangular array is like a simple one-dimensional array. The operators defined for this one-dimensional array make it look like a multi-dimensional array (But we know that it really isn't). Is this rectangular array that we call a matrix, just like the multi-dimensional arrays in computers? I mean, is it true that we say "$A_{ij}=A_{m\times (i-1)+j}$" ? ( Notice that in this way, the matrix $A$ would be equal to $(A_1,A_2,\dots,A_{{n^2}})$ )

Another way of defining a matrix more precisely might be this one :
$A \in M_{m,n}(F)=\{(A_1,A_2,\dots,A_m):\forall i \space A_i \in F^n\}$

Is any of these two definitions correct? If not, please suggest a better definition.

Note : My point is to clarify the definition in my mind.

An $(m \times n)$-matrix with entries in $F$ is simply a function $M\colon \{1, \dotsc, m\} \times \{1, \dotsc, n\} \to F$.