My preferred take on the Weil pairing is via Mumford's theta group, which is a group scheme $\mathcal{G}$ fitting into a short exact sequence

$1 \rightarrow \mathbb{G}_m \rightarrow \mathcal{G} \rightarrow E[n] \rightarrow 0$.

Note that the theta group itself is a noncommutative central extension of one commutative group (scheme) by another.  In particular, if you take $P_1, P_2 \in E[n]$ (I really mean $T$-valued points for some $K$-scheme $T$...) then (i) lift to $\tilde{P}_1, \tilde{P}_2$ in $\mathcal{G}$, and (ii) form the commutator $e(P_1,P_2) = [\tilde{P}_1,\tilde{P_2}]$, then since this maps to the commutator $[P_1,P_2]$ 
in the commutative group $E[n]$, i.e., it maps trivially and therefore lives in $\mathbb{G}_m$.  Moreover, 
since $\mathbb{G}_m$ is central, this element $e(P_1,P_2)$ is independent of the choice of lifts.  It is also not too hard to check that it lands in $\mu_n$ (the nth roots of unity) inside $\mathbb{G}_m$ and in fact that the map $e: E[n] \times E[n] \rightarrow \mu_n$ is nondegenerate: i.e., it puts $E[n]$ into self Cartier duality.  For all this, see Mumford's book *Abelian Varieties*.

Indeed one of the advantages of this approach is that it generalizes very gracefully to the setting of a polarized abelian variety $(A,L)$.  

This take on the Weil pairing has been vitally useful to me in my research in the Galois cohomology of abelian varieties: see for instance $\S 6$ of [this paper][1] where theta groups are studied in a more general Galois cohomological context.  (I am not the only one or even the first to have studied such things: see especially the 2002 paper of Polishchuk that appears in the bibliography.) 

[1]: http://math.uga.edu/~pete/wc2.pdf