Let $E$ be an elliptic curve over a field $K$ and $n$ be an integer such that the characteristic of the field does not divide $n$. Let $e_n: E[n] \times E[n] \rightarrow \mu_n$ be the Weil pairing and $P$ a point of order $n$. Show that there is an integer $k$ such that $e_n(P, Q) = 1$ if and only if $Q = kP$.
This in an exercise in a book. One side of the proof is easy: if $Q = kP$ then
$$\begin{split} e_n(P, Q) &= e_n(P, kP)\\ &= e_n(P,P)^k\\ &= 1 \end{split} $$
no matter what $k$ we choose.
I'm having problems with the other implication. My reasoning is the following: since $P$ has order $n$ it must be the generator of one of the cycles in $E[n] = \mathbb{Z}_n \times \mathbb{Z}_n$. Then we have a base $\{P, R\}$ for $E[n]$ and we can write $Q = c_1 P + c_2 R$. Thus,
$$\begin{split} e_n(P,Q) &= e_n(P,c_1 P + c_2 R)\\ &= e_n(P,P)^{c_1} \cdot e_n(P, R)^{c_2}\\ &=e_n(P, R)^{c_2} \end{split}$$
So $e_n(P,Q)$ may be $1$ even if the $Q$ is not even a multiple of $P$ depending on the value of $e_n(P,R)$. What am i missing?