When you make $m$ times a sample of $n$ i.i.d. random variables, then you have generated in all $m\cdot n$ i.i.d. random variables. All these $m\cdot n$ variables are still independent of each other, the computer takes care of that. We could just number them from $X_1$ to $X_{mn}$.
Now you have stored them row by row in a matrix, so row 1 consists of $X_1,..., X_n$, row 2 consists of $X_{n+1},...,X_{n+n}$, row 3 of $X_{2n+1},...,X_{2n+n}$ and so on, and row $m$ of $X_{(m-1)n+1},...,X_{mn}$. If you look at any column, for example the first column $(X_1, X_{n+1},\dots, X_{(m-1)n +1})$, or the $j$-th, $(X_j, X_{n+j},\dots, X_{(m-1)n +j})$, the variables are still independent of each other.
This also holds when the simulated $X_1, \dots, X_{mn}$ have a distribution that is not normal. You only need that they are independent :-)
In many texts about linear regression, one writes $$ \mathbf{Y} = \beta_0 + \beta_1 \mathbf{X} + \epsilon $$ and in reality means $$ \begin{array}{1} Y_1\\ \vdots\\Y_n \end{array} \begin{array}{1} =\\ \ \\= \end{array} \begin{array}{1} \beta_0 +\beta_1 X_1 +\epsilon_1\\\qquad \vdots\\ \beta_0 +\beta_1 X_n +\epsilon_n \end{array}. $$