Definition of extended euclidean space

Question

My professor told me that the definition of the extended euclidean space (which in my course is denoted by $\mathbb{E}_n^{\star}$) is “needed” to define a projective space, but I can’t see how that could be done. Can’t we define the extended euclidean space as $\mathbb{E}_n^{\star}:=P(\mathbb{E}_{n+1})\cup\Omega$, where $P(V)$ is a projective space over some field and $\Omega$ is the hyperplane at infinity? Am I missing something?