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 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.)
Added: Hilbert symbols show up in my paper too, to say the least. From the cohomological perspective this is hardly a surprise, since the Hilbert symbol is really the cup product $H^1(K,\mu_n) \times H^1(K,\mu_n) \rightarrow H^1(K,\mu_n^{\otimes 2})$ in the case where $\mu_n \cong \mathbb{Z}/n\mathbb{Z}$, whereas $E[n] \cong \mu_n \times \mu_n \cong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n \mathbb{Z}$ when there is full $n$-torsion over $K$. The Weil pairing is defined without any rationality assumptions on the $n$-torsion but becomes a lot harder to work with explicitly when the Galois module structure on $E[n]$ is nontrivial.
