Prove that invertible metrices set is an open set in a given space, and the determinant is continuous

Prove that invertible metrices set is an open set in a given space, and
the determinant is continuous

Given a matrix $M_{n\times m}$, we can think about it as a vector in
$\mathbb{R}^{n\times m}$ (How come?).
How can I prove that the set of all the invertible metrices of size
$n\times n$ is an open set in $\mathbb{R}^{n^2}$, and that the function
$f:M\mapsto M^{-1}$ is continuous?
I understand that $f$ is the determinant of $M$, and $f = \det(M)\ne 0$,
and becuase $f$ is a linear function, it's a continuous function. But I'm
not 100% sure why $f$ is a linear.
The invertible metrices set is $\mathbb{R}^{n^2}\setminus\{0\}$ (am I
wrong?), and that's why the set is an open set?
Thank you!