A necessary (not sufficient) condition for two vectors $u,v$ in $\mathbb{R}^2$ to sum up to zero is that they have to be linearly dependent, i.e.
\begin{equation}
\det\left(u,v\right) = 0
\end{equation}
Is there a similar necessary condition for $n$ vectors $u\_1,u\_2\dots u\_n$ in $\mathbb{R}^2$ ?

I believe the following could be a candidate but it's rather ugly and does not have nice symmetries:
\begin{equation}
\det\left(u\_1,u\_2\right) +\det\left(u\_1,u\_3\right) + \cdots + \det\left(u\_1,u\_n\right) = 0
\end{equation}
Any better solutions?