Can I wipe a MacBook Air SSD without having a recovery partition and still be able to do a new installation of OS X? I tried to update my old MacBook Air from OS X 10.8 to the newest El Capitan. However, now (directly after the installation) when I boot it up the loading bar will stop at 100% and nothing happens.

Then I wanted to reinstall OS X (10.8) but it says that it can't install an older version of OS X. 

Now I'm really tempted to wipe the SSD but I'm worried that if I completely wipe the SSD it won't be able to do anything with the MacBook since it doesn't have a recovery partition.

Is it safe to wipe the SSD?
If your Mac shipped with OS X Lion or later, you'll be able to use Internet Recovery - in which case it's safe to fully erase your SSD, as your Mac is capable of fetching the recovery partition over the Internet.

You can enter Internet Recovery by holding Cmd+R during boot.