Complete linear hypergraph on $n$ points

Question

A complete linear hypergraph is a hypergraph $H=(V,E)$ such that if $e_1, e_2\in E$ then $|e_1\cap e_2|=1$.

Let $n\geq 3$. Can we pick $E\subseteq {\cal P}(\{1,\ldots, n\})$ such that

1. $(\{1,\ldots, n\}, E)$ is a complete linear hypergraph, and
2. $|E| = n+1$

?