Answer
Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- The Prophet (peace and blessings be upon him) was reported to have wiped over his socks.
2- We are allowed to wipe over the socks as he allowed us to do that for a whole day and for three days in case of traveling.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and an Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states:
It is not true that the Prophet (peace and blessings be upon him) always removed the socks while performing ablution (wudu’). Rather, he often wiped over them.
Imam Al-Bukhari (may Allah have mercy on him) reports on the authority of Al-Mughirah ibn Shu`bah that he said, “I was with the Prophet (peace and blessings be upon him) during a journey and I volunteered to remove his leather socks (when he was going to make ablution). He stopped me, saying, ‘Leave them, for I wore them while I was in a state of purity’ and he wiped over them.”
Furthermore, we learn from the Companions that the Prophet (peace and blessings be upon him) allowed us to wipe over socks for a whole day and for three days while travelling, as long we do not take them off or do anything to make ghusl (ritual bath) obligatory.
Allah Almighty knows best.