  We put forward a simple construction of genuinely entangled subspaces --
subspaces supporting only genuinely multipartite entangled states -- of any
permissible dimensionality for any number of parties and local dimensions. The
method uses nonorthogonal product bases, which are built from totally
nonsingular matrices with a certain structure. We give an explicit basis for
the constructed subspaces. An immediate consequence of our result is the
possibility of constructing in the general multiparty scenario genuinely
multiparty entangled mixed states with ranks up to the maximal dimension of a
genuinely entangled subspace.
