pistolp,
Unfortunately, hyperextension of the knee can cause several different injuries.
The cruciate ligaments are at risk, the ACL and PCL. If you torn one of them, the knee usually swells up over night, and feels like a water ballon. The swelling can even extend up around the top of the patella, like a horseshoe. And of course, due to the swelling, bending the knee would feel like bending a hot dog.
The meniscal cartilages are also in danger of being pinched in this type of mechanism. If the meniscus tears at its attachment to the capsule, it may bleed and produce swelling in the joint. But, tears in the substance of the cartilage usually don't cause swelling right off the bat. This type usually causes locking with activity and much later, may cause an inflammatory type swelling.
It is possible to avulse the plantaris tendon from its attachment in the back of the knee. This is a small muscle with a long tendon, that as upright walker, humans do not use anymore. It is left over from a time when we had grasping toes. Now days, it is used as tendon grafts when needed. But, I digress. The plantaris can rupture of get pulled off, and it will cause pain deep in the posterior knee and calf muscles. There is usually swelling there, with bruising in the back of the knee and calf.
If you twisted when you went down, it is also possible to sublux or dislocate the patella. This would cause swelling around the patella, with tenderness usually on the medial side of the patella. Occasionally, there will be some bruising around the patella.
It is also possible to have a small compression fracture (like a bone bruise) of the tibial plateau. This can cause swelling in the joint. This is usually only seen on an MRI.
Or you could have had a generalized capsular injury. Which is tender, may have some swelling and stiffness, but on exam after the swelling goes down, the knee is okay.
Like you can see there are a lot of things that can be injured in the knee. The only way to know for sure, would be to have someone examine the knee. In the mean time, use RICE - Rest, ice, compression, and elevation. If you can take them, some tylenol or antiinflammatory medicines may help with the discomfort. Try not to let the knee lose range of motion. Gentle range of motion exercises, within limits of pain, will help. Try to elevate the leg as a straight line, as a unit. Do not drape the knee over a pillow (even if it feels better than way), because that can cause a flexion contracture, which are very hard to get rid of.
Again, if the knee is not feeling better soon, or you develop other symptoms (locking, wobbly feeling, instability, etc), you might want to have the knee examined by an orthopedic surgeon.
Good luck.