  We present a different version of the well-known connection between
Hopf--Galois structures and skew braces, building on a recent paper of A. Koch
and P. J. Truman. We show that the known results that involve this connection
easily carry over to this new perspective, and that new ones naturally appear.
As an application, we present new insights on the study of the surjectivity of
the Hopf--Galois correspondence, explaining in more detail the role of bi-skew
braces in Hopf--Galois theory.
