  We present a novel framework to decompose three-nucleon forces in a momentum
space partial-wave basis. The new approach is computationally much more
efficient than previous methods and opens the way to ab initio studies of
few-nucleon scattering processes, nuclei and nuclear matter based on
higher-order chiral 3N forces. We use the new framework to calculate matrix
elements of chiral three-nucleon forces at N2LO and N3LO in large basis spaces
and carry out benchmark calculations for neutron matter and symmetric nuclear
matter. We also study the size of the individual three-nucleon force
contributions for $^3$H. For nonlocal regulators, we find that the sub-leading
terms, which have been neglected in most calculations so far, provide important
contributions. All matrix elements are calculated and stored in a user-friendly
way, such that values of low-energy constants as well as the form of regulator
functions can be chosen freely.
