  We study conformal symmetry breaking differential operators which map
differential forms on $\mathbb{R}^n$ to differential forms on a codimension one
subspace $\mathbb{R}^{n-1}$. These operators are equivariant with respect to
the conformal Lie algebra of the subspace $\mathbb{R}^{n-1}$. They correspond
to homomorphisms of generalized Verma modules for ${\mathfrak so}(n,1)$ into
generalized Verma modules for ${\mathfrak so}(n+1,1)$ both being induced from
fundamental form representations of a parabolic subalgebra. We apply the
F-method to derive explicit formulas for such homomorphisms. In particular, we
find explicit formulas for the generators of the intertwining operators of the
related branching problems restricting generalized Verma modules for
${\mathfrak so}(n+1,1)$ to ${\mathfrak so}(n,1)$. As consequences, we find
closed formulas for all conformal symmetry breaking differential operators in
terms of the first-order operators $d$, $\delta$, $\bar{d}$ and $\bar{\delta}$
and certain hypergeometric polynomials. A dominant role in these studies will
be played by two infinite sequences of symmetry breaking differential operators
which depend on a complex parameter $\lambda$. These will be termed the
conformal first and second type symmetry breaking operators. Their values at
special values of $\lambda$ appear as factors in two systems of factorization
identities which involve the Branson-Gover operators of the Euclidean metrics
on $\mathbb{R}^n$ and $\mathbb{R}^{n-1}$ and the operators $d$, $\delta$,
$\bar{d}$ and $\bar{\delta}$ as factors, respectively. Moreover, they are shown
to naturally recover the gauge companion and $Q$-curvature operators of the
Euclidean metric on the subspace $\mathbb{R}^{n-1}$, respectively.
