Embedding $\mathbb{CP}^n$ into $\mathbb{R}^{4n-1}$

Question

I'm looking for a generalization of the map of $\mathbb{C}P^1$ into $\mathbb{R}^3$ as a sphere. Are there any such (preferably canonical) embeddings of $\mathbb{C}P^n$ into $\mathbb{R}^{4n-1}$? (And is $4n-1$ the correct dimension to be asking this question in?)