Expectation of a matrix of random variables

Question

Let $A_{ij}$ be a collection of $N^2$ random variables where $i,j\in\{1,\dots,N\}$. Let $f(A_{ij})$ denote the joint probability distribution of random variables $A_{ij}$. How do we define the "expected matrix" for this?. For instance, for a scalar random variable $x$ with pdf as $f(x)$, one has the usual definition of expectation as $\int xf(x)dx$. How do we define such a notion for matrices?