Suppose $\Omega=(0,1)$, $\mathcal{B}$ is the Borel $\sigma$-algebra on $(0,1)$ and $P$ the Lebesgue measure. Let $\Phi:\mathbb{R}\to(0,1)$ denote the standard normal CDF. Then, we can generate $X\sim N(0,1)$ as follows: $X(\omega)=\Phi^{-1}(\omega)$. This works because $$ P(\omega:X(\omega)\leq c)=P(\omega:\Phi^{-1}(\omega)\leq c)=P(\omega:\omega\leq\Phi(c))=\Phi(c). $$ My question: how do I generalize this to generate $X_1,\ldots,X_n$ $\sim$ i.i.d. $N(0,1)$? I would like to do this with a single $(\Omega,\mathcal{B},P)$, possibly redefined from their above meanings.
I'm guessing that $\Omega=(0,1)^n$, $\mathcal{B}$ is some sort of product of the one-dimensional Borel $\sigma$-algebra. But I lack the language to describe this precisely. I think this is a standard exercise so I would really appreciate a rigorous work out so I can learn this properly. A reference recommendation for this type of constructions will also be very helpful.