  Banagl's method of intersection spaces allows to modify certain types of
stratified pseudomanifolds near the singular set in such a way that the
rational Betti numbers of the modified spaces satisfy generalized Poincar\'{e}
duality in analogy with Goresky-MacPherson's intersection homology. In the case
of one isolated singularity, we show that the duality isomorphism comes from a
nondegenerate intersection pairing which depends on the choice of a chain
representative of the fundamental class of the regular stratum. On the
technical side, we use piecewise linear polynomial differential forms due to
Sullivan to define a suitable commutative cochain algebra model for
intersection spaces. Our construction parallels Banagl's commutative cochain
algebra of smooth differential forms modeling intersection space cohomology,
and we show that both algebras are weakly equivalent.
