  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.
