This asus has a Broadcom BCM4313 wifi card. The linux kernel that ships with Slackware 13.37 comes with an open-source driver for this wifi card. Unfortunately, it does not come with the firmware for the card, so the driver is useless. If you try to download and install the driver from Broadcom it crashes the machine (even if you blacklist every other module).
The solution is to download and install the firmware according to the driver's README (which you can find at /usr/src/linux-18.104.22.168/drivers/staging/brcm80211/README). Once the drivers are copied to /lib/firmware/brcm/ you need to make symlinks "bcm43xx-0.fw" and "bcm43xx_hdr-0.fw" to the files which look closest like those.
Of course the git repository that has the files is not working, so you have to pull them from somewhere else. You can get the archive from debian here: http://packages.debian.org/sid/firmware-brcm80211 (download the source package's .tar.gz file and extract it)
Once you set up the firmware, just reboot and the machine should attempt to load the brcm80211 module and the right firmware automatically. Don't use the "wl" driver from broadcom as it will crash the machine. Add "wl", "b43" and "ssb" to the /etc/modprobe.d/blacklist.conf file just in case it tries to load those.
Apparently, that stuff doesn't work the way it should. You have to upgrade to the latest 2.6 kernel (22.214.171.124 as of this writing) and load the 'brcmsmac' driver, as this is the new driver used by the BCM4313 on the latest 2.6 kernels. Blacklist all the other drivers first ('wl', 'brcm80211', 'b43', 'ssb', 'b43-legacy', 'bcma'). I'm not sure if this is because the latest firmware is incompatible with older driver versions, but it kept crashing my machine to do anything but use the latest kernel and the brcmsmac driver with the latest firmware. What a pain in the ass.