Chrome and Skype takes up a lot of storage
You cannot do anything about those two monsters using so much data – they're known for that. But still, there's something to work around that, enabling you to "update" without losing your data:


Backup for safety (data only)

adb backup -noapk -f Skype.ab com.raider.skype
adb backup -noapk -f Chrome.ab com.android.chrome

Remove the apps, but keep their data

adb uninstall -k com.raider.skype
adb uninstall -k com.android.chrome

Install the apps anew using e.g. the Playstore app


If that doesn't work out (as the amount of data still is too much), ommit the -k in the second step – and to restore your data, follow the third step by

adb restore Skype.ab
adb restore Chrome.ab


This requires your computer to be equipped with adb (see the adb tag-wiki for details, including a minimal install as you won't need the entire SDK for that).