How can I extend a Wireless Network WIRELESSLY?

Yes, this is possible but it depends on the specifics of your wireless router. You would need to setup a second device as a wireless bridge. In some configuration pages this is called WDS mode or wireless access point w/ WDS.

First, check to see if your current router supports this, then your best bet for compatibility would be to purchase an identical router, or a WAP by the same manufacturer, as the second device. I hope this helps.

Just kidding. Ha!
I don't see why a WAP wouldn't work...
Just follow the destructions

You will need a router that offers the bridging capability, most do not. If you can run a wire to somewhere on the second floor, you can use any standard router with the following configuration.

Connecting two (or more) SOHO broadband routers together.

Configure the IP address of the secondary router(s) to be in the same subnet as the primary router, but out of the range of the DHCP server in the primary router. For instance DHCP server addresses 192.168.0.2 through 192.168.0.100, I would assign the secondary router 192.168.0.254 as it's IP address, 192.168.0.253 for another router, etc.

Note: Do this first, as you will have to reboot the computer to connect to the router again for the remaining changes.

Disable the "DHCP" server in the secondary router.

Setup the wireless section just the way you would if it was the primary router, channels, encryption, etc.

Connect from the primary routers LAN port to one of the LAN ports on the secondary router. If there is no uplink port and neither of the routers have auto-sensing ports, use a cross-over cable. Leave the WAN port unconnected!

This procedure bypasses the routing function (NAT layer) and configures the router as a switch (or wireless access point for wireless routers).

Hope all goes well.