The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to $Hom_{Chow\otimes \mathbb Q}(M(X),M(\mathbb P^n))$ for $n\ge \dim X$. Is there a conceptual explanation for the (composite) isomorphism obtained (i.e. of the expression of $K_0$ in terms of cycles on $X\times \mathbb P^n$)?

Certainly, the sides of this isomorphism are related somehow; yet I don't know whether 'standard' arguments can identify them directly without mentioning the sum of Chow groups.