  We develop a formal construction of a pointwise divergence-free basis in the
nonconforming virtual element method of arbitrary order for the Stokes problem
introduced in [19]. The proposed construction can be seen as a generalization
of the divergence-free basis in Crouzeix-Raviart finite element space [10, 17]
to the virtual element space. Using the divergence-free basis obtained from our
construction, we can eliminate the pressure variable from the mixed system and
obtain a symmetric positive definite system. Several numerical tests are
presented to confirm the efficiency and the accuracy of our construction.
