Skew braces of size $pq$

We construct all skew braces of size $pq$ (where $p>q$ are primes) by using Byott's classification of Hopf--Galois extensions of the same degree. For $p\not\equiv 1 \pmod{q}$ there exists only one skew brace which is the trivial one. When $p\equiv 1 \pmod{q}$, we have $2q+2$ skew braces, two of which are of cyclic type (so, contained in Rump's classification) and $2q$ of non-abelian type.