  We introduce a one-parameter family of random infinite quadrangulations of
the half-plane, which we call the uniform infinite half-planar quadrangulations
with skewness (UIHPQ$_p$ for short, with $p\in[0,1/2]$ measuring the skewness).
They interpolate between Kesten's tree corresponding to $p=0$ and the usual
UIHPQ with a general boundary corresponding to $p=1/2$. As we make precise,
these models arise as local limits of uniform quadrangulations with a boundary
when their volume and perimeter grow in a properly fine-tuned way, and they
represent all local limits of (sub)critical Boltzmann quadrangulations whose
perimeter tend to infinity. Our main result shows that the family
(UIHPQ$_p$)$_p$ approximates the Brownian half-planes BHP$_\theta$, $\theta\geq
0$, recently introduced in Baur, Miermont, and Ray (2016). For $p<1/2$, we give
a description of the UIHPQ$_p$ in terms of a looptree associated to a critical
two-type Galton-Watson tree conditioned to survive.
