What is the notation for the set of all $m\times n$ matrices?

Given that $\mathbb{R}^n$ is the notation used for n-dimensional vectors, is there an accepted equivalent notation for matrices?

If $A$ is an $m\times n$ matrix, then $$A \in \mathbb{R}^{m\times n}$$

As you can see from the previous two answers, several notations are in common usage, so it's best to say what you mean the first time you use it to be completely clear. (For what it's worth, I often use $M_{m\times n}(\mathbb{R})$, which is different again).

The vector space of real matrices with $n$ rows and $m$ columns is denoted by $\mathcal{M}_{n,m}(\mathbb R)$ and its $nm$-dimensional vector space so it's isomorphic to $\mathbb R^{nm}$

The notation $A\in\mathbb{R}^{m\times n}$ is in fact correct. We should be careful with the symbol '$\times$' that in this case does not means an integer product but a group product that, as a result, is a new group in $\mathbb{R}^2$ with elements of the form $(m, n)$, and thus the matrix $A$ is contained in it.