I need to compute $\chi(\mathbb{C}\mathrm{P}^2)$ using techniques from differential topology. I cannot think of any theorems that are particularly useful for this computation, so I think that I will have to find a vector field on $\mathbb{C}\mathrm{P}^2$ with isolated zeros and compute the index of the vector field about these zeros. My first idea to find a vector field with isolated zeros was to recall the diffeomorphism $\mathbb{C}\mathrm{P}^2 \cong S^5/\sim$ where $(z^1,z^2,z^3) \sim (u^1,u^2,u^3)$ if and only if there exists $w \in S^1$ with $(z^1,z^2,z^3) = (wu^1,wu^2,wu^3)$. Then it would suffice to find a vector field on $S^5$ which descends to a vector field on $S^5/\sim$ with isolated zeros. However, I have had some difficulties making this approach work, so I was hoping that someone could help me out here.
Edit 1: I should add that while I am limited to the tools of differential topology for this problem, I do not have to follow the outline I have thus far; that is, I can find a vector field on $\mathbb{C}\mathrm{P}^2$ with isolated zeros and compute its Euler characteristic from there in anyway (the vector field does not have to come from $S^5$).
Edit 2: I am also not limited to computing the Euler characteristic directly from a vector field with isolated zeros. For example, I can use things like the Gauss mapping too.