Macbook Air SSD Problem after erase

I was trying to fresh-install my os, so i started on recovery (pressed command+r) and I erased my drive. Than after reboot i click the option "reinstall osx", but when it gets to the point I need to choose where to install osx, i don't have any options. so i went to disk utility, i clicked "verify disk" and "check disk" on my main drive (the ssd drive) and it says everything is ok. The weird thing is that i only have two tabs: 'first aid' and 'partition'. on partition i cant do enything. it looks like it is not partitioned and it is not ready for startup, because also when i click 'choose start up disk' i dont have any option, blank.
when im on disk utility there is also another weird thing: I see it only on one line:

that how it looks when it is normal (on my macbook pro):

i think it is locked or something.
any one has any idea?
thanks ahead, and sorry for the English. i'm not local

Highlight the internal drive in the side bar

Go to the Partition tab

Under Patition Layout, click the dropdown and select 1 Partition

Name the partition to what you would like, click Apply - it will also format it.

Thing is - I can't click nothing on the partition tab!!
i made this video to make it more clear: https://youtu.be/A35VVK_uaeI
https://youtu.be/A35VVK_uaeI

Please guy, there must be something i can do

I have seen a few problems with the Logical Volume Group (LVG) - but not that one - not really sure how to get around where you are.

Two suggestions - the first is take it to an Apple store and ask them to install OS X for you.

No store close, what I would try will require an OS X bootable installer on a thumb drive.
I'd boot to it and start off by erasing disk 1.
That may or may not permit you to partition the drive.

I have had one problem that I could only solve by getting rid of the LVG.
Here is one tutorial for doing that - link.

Ahh the joys of the thumb drive installer strikes yet again!!

Or it could be because the 2011 macbook air had a firmware update to prevent this problem with the drive lock and file corruption issues.

apple did release important information on it here is the link.

https://www.apple.com/support/macbookair-flashdrive/

and most likely you have a 2012 model because the 2011 used a peta micro drive.