#Japt, 8 bytes

Returns an array of lines.

õ ¬å+ zU

Try it

Saved 2 bytes thanks to Kevin.

õ ¬å+ zU     :Implicit input of integer U
õ            :Range [1,U]
  ¬          :Join to a string
   å+        :Cumulatively reduce by concatenation
      zU     :Rotate clockwise by 90 degrees U times

Japt, 8 bytes

Returns an array of lines.

õ ¬å+ zU

Try it

Saved 2 bytes thanks to Kevin.

õ ¬å+ zU     :Implicit input of integer U
õ            :Range [1,U]
  ¬          :Join to a string
   å+        :Cumulatively reduce by concatenation
      zU     :Rotate clockwise by 90 degrees U times

#Japt -R, 10 bytes

õ ¬å+ ú zU

Try it

õ ¬å+ ú zU     :Implicit input of integer U
õ              :Range [1,U]
  ¬            :Join to a string
   å+          :Cumulatively reduce by concatenation
      ú        :Right pad each with spaces to the length of the longest
        zU     :Rotate clockwise by 90 degrees U times
               :Implicitly join with newlines & output

#Japt, 8 bytes

Returns an array of lines.

õ ¬å+ zU

Try it

Saved 2 bytes thanks to Kevin.

õ ¬å+ zU     :Implicit input of integer U
õ            :Range [1,U]
  ¬          :Join to a string
   å+        :Cumulatively reduce by concatenation
      zU     :Rotate clockwise by 90 degrees U times

#Japt -R, 12 bytes

õ ¬å+ ú zUu4

Try it

õ ¬å+ ú zUu4     :Implicit input of integer U
õ                :Range [1,U]
  ¬              :Join to a string
   å+            :Cumulatively reduce by concatenation
      ú          :Right pad each with spaces to the length of the longest
        z        :Rotate clockwise by 90 degrees
         Uu4     :  U modulo 4 times
                 :Implicitly join with newlines & output

#Japt -R, 10 bytes

õ ¬å+ ú zU

Try it

õ ¬å+ ú zU     :Implicit input of integer U
õ              :Range [1,U]
  ¬            :Join to a string
   å+          :Cumulatively reduce by concatenation
      ú        :Right pad each with spaces to the length of the longest
        zU     :Rotate clockwise by 90 degrees U times
               :Implicitly join with newlines & output