Intersection pairing in homology

Question

Let $M$ be an $n$-dimensional closed connected manifold and suppose it is $k$-oriented where $k$ is a field. In what follows, all the coefficients are supposed to be in $k$. Define a pairing $$\cdot\colon H_{k}(M)\times H_{n-k}(M)\rightarrow k$$ given by $a\cdot b = \langle P_D(a), b\rangle = \langle P_D(a)\cup P_D(b),[M]\rangle$ where $P_D$ is the Poincaré duality isomorphism and $[M]$ a $k$-fundamental class. The pairing is clearly non-degenerate. I would like to use this pairing to show some properties concerning the diagonal in $M\times M$. Let$[\Delta]$ denote the fundamental class of the diagonal in $M\times M$. Take a basis $\{a_i\}$ of $H_{*}(M)$ and $\{b_i\}$ the dual basis with respect to $\cdot$, i.e. $a_i \cdot b_j =\delta_{ij}$. I want to show that $[\Delta] = \sum_{i} a_i \times b_i$, but I don't know how to proceed.