Cees, I don't FSX but this problem sounds very familiar in the older sims.
The Skyhawk (if accurately modeled) has an unstable landing gear footprint.
Imagine that you're looking down at the plane and you can see the three tires.
It looks like a triangle.
In this case, the triangle will look like its tall at the top and the base will be narrow.
The Skyhawk also has a relatively high COG which doesn't help.
Imagine sitting on a three legged stool with the same triangle at its base.
To fix it, or at least make it land-able, you'll need to edit the [contact points] section of the aircraft.cfg file.
Check first that the two main gear legs have the same length (Z axis). In some cases they won't.
Next, try making that base of the triangle wider (Y axis). You don't need to add a lot, try adding a half of a foot to each main gear. One will be positive, the other will be negative.
You can test these changes at a normal airport. Just taxi around like you've had a few too many to drink.
Be aggressive, make sharp turns, jam on the brakes in the middle of a turn, do all the stuff you normally
wouldn't do.
You could also make the triangle a little more stable by adding a second nose wheel. Offset the original nose wheel to the right by about a half of a foot. Add the new wheel and offset it the same amount in a negative direction so the new nose wheels act like bogies.
Another thing to check is the damping and damping ratio for the gear. If they are set too high the shock absorbers will act like catapults.
If you don't have a lot of experience with these edits, copy the [contact points] section and paste it in this thread and I'll show you what I mean.