Calculation of Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP}^{n+1}$

Question

I would like to go about finding an explicit representative of the Poincare dual to $\mathbb{CP}^n$ in $\mathbb{CP^{n+1}}$, I am following Bott & Tu and would like an explicit form to be wedged with itself and integrated to give the Chern numbers for $\mathbb{CP^{n+1}}$. The reason I am having difficulties is that, although I have managed to calculate the Poincare dual for simple things like circles in the torus, I can't think of how to go about choosing the right coordinates to work in. Should I for example consider $\mathbb{CP}^n =\{[z_0,\ldots,z_n,0]\}$ and then look for the Poincare dual to this submanifold in standard coordinates over the patches $\{U_i=\{[z_0,\ldots,z_{n+1}]\colon z_i\neq0\}\colon i=1,\ldots,n\}$ before patching these together with a partition of unity? The problem with this idea seems to me that I won't end up with a global form, it won't be defined at the point $[0,\ldots,0,1]$. Moreover, given any copy of $\mathbb{CP}^n$ in $\mathbb{CP^{n+1}}$, there doesn't seem to be any way to cover $\mathbb{CP}^{n+1}$ with $\mathbb{CP}^n$ contained in every patch.