  OMC-2/3 is one of the nearest embedded cluster-forming region that includes
intermediate-mass protostars at early stages of evolution. A previous CO (3--2)
mapping survey towards this region revealed outflow activity related to sources
at different evolutionary phases. The present work aims to study the warm gas
in the high-velocity emission from several outflows found in CO (3--2) emission
by previous observations, determine their physical conditions, and make
comparison with previous results in low-mass star-forming regions. We use the
CHAMP+ heterodyne array on the APEX telescope to map the CO (6--5) and CO
(7--6) emission in the OMC-2 FIR 6 and OMC-3 MMS 1-6 regions, and to observe
$^{13}$CO (6--5) at selected positions. We analyze these data together with
previous CO (3--2) observations. In addition, we mapped the SiO (5--4) emission
in OMC-2 FIR 6. The CO (6--5) emission was detected in most of the outflow
lobes in the mapped regions, while the CO (7--6) was found mostly from the
OMC-3 outflows. In the OMC-3 MMS 5 outflow, a previously undetected extremely
high velocity gas was found in CO (6--5). This extremely high velocity emission
arises from the regions close to the central object MMS 5. Radiative transfer
models revealed that the high-velocity gas from MMS 5 outflow consists of gas
with $n_{\rm H_2}=$10$^4$--10$^{5}$ cm$^{-3}$ and $T>$ 200 K, similar to what
is observed in young Class 0 low-mass protostars. For the other outflows,
values of $n_{\rm H_2}>$10$^{4}$ cm$^{-3}$ were found. The physical conditions
and kinematic properties of the young intermediate-mass outflows presented here
are similar to those found in outflows from Class 0 low-mass objects. Due to
their excitation requirements, mid$-J$ CO lines are good tracers of extremely
high velocity gas in young outflows likely related to jets.
