Let $(a_{ij})_{1 \le i,j \le n}$ be a real orthogonal matrix. Show that $\left| \sum_{1 \le i,j \le n} a_{ij}\right| \le n.$
Naively applying the Cauchy-Schwarz inequality only gives $n^{\frac{3}{2}}$ (but only relies on the columns being of norm $1$, and not orthogonality). How do we get the stronger bound $n$?