Do Octonionic Projective spaces exist or defined similar to $\Bbb RP^n$, $\Bbb CP^n$, $\Bbb HP^n$? If so, are they symmetric spaces?

I am asking this question because I've never seen Octonionic Projective spaces as examples of the Compact Rank One Symmetric Spaces or the manifolds of positive sectional curvature except its 2-dimensional case i.e. Cayley plane $\Bbb OP^2$ (or $\Bbb{Ca}P^2$). What can be say about $\Bbb OP^n$ (or $\Bbb{Ca}P^n$)?

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**Update:** Please give a proof or a reference for non-existence of Octonionic Projective spaces for $n\geq 3$.

Related question: [Quaternion Projective Space $\mathbb HP^n$ and Octonionic Projective Space $\mathbb OP^n$](https://math.stackexchange.com/questions/600772)