String Steps

I also looked at the String Steps problem.
I believe that it is possible to arrange binary strings in this way, given that the first string equates to zero (all 2^n digits are 0) and the last string is all zeroes except for the very first digit. Beyond n = 3 it gets a bit complicated, but I believe that if it is possible for n= 3 at least, it is possible for all positive n >= 1.
n =1
0
1
n = 2
00
01
11
10
n = 3
000
001
011
010
110
111
101
100