Prophet (SAW)'s Prayers

At the begining of making wudu
بِسْمِ اللَّهِ الرَّحْمَنِ الرَّحِيمِ
"I begin in the name of Allah, who is the Most Merciful, Most Gracious".
It is stated in the hadith that one should begin wudhu with Allah's name. No words have been specified and our elders read the above. It is recorded in some ahadith that wudhu is valid although no "Bismillah" is recited.