Let $W\subset \mathbb P^n$ be a smooth projective variety of codimension 2 and let $X$ be the blow-up of $\mathbb P^n$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $A^2(X)=A^2(\mathbb P^n)\oplus A^1(W)$, so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ is given by $H^2$ and line bundles on $W$. However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in this basis?