About the definiton of a Matrix

Question

I have a basic question about the definition of a matrix. It is usually defined as a rectangular or two-dimensional array which is OK. Could we define a matrix as co-domain of a function form $\mathbb{N}_k\times\mathbb{N}_{k'}$ to a subset of $\mathbb{R}$ where $\mathbb{N}_k$ stands for $$\{1,\ldots,k\}$$

Thanks!