The van Kampen–Flores theorem states that the n-skeleton of a (2 n + 2) -simplex does not embed into R 2 n . We give two proofs for its generalization to a continuous map from a skeleton of a certain regular CW complex (e.g. a simplicial sphere) into a Euclidean space. We will also generalize Frick and Harrison's result on the chirality of embeddings of the n-skeleton of a (2 n + 2) -simplex into R 2 n + 1 .