Let $\mathbb K$ be a field of characteristic 0. Given $n$ linear forms in $R=\mathbb K[x_1,\ldots,x_k]$, with no two proportional, in one of our main results we show that the ideal $I\subset R$ generated by all $(n-2)$-fold products of these linear forms has linear graded free resolution. This result helps determining a complete set of generators of the symmetric ideal of $I$. Via Sylvester forms we can analyze from a different perspective the generators of the presentation ideal of the Orlik-Terao algebra of the second order; this is the algebra generated by the reciprocals of the products of any two (distinct) of the linear forms considered. We also show that when $k=2$, and when the collection of $n$ linear forms may contain proportional linear forms, for any $1\leq a\leq n$, the ideal generated by $a$-fold products of these linear forms has linear graded free resolution.