In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space ${ℝ}^n$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a*b=(a+b){(1-ab)}^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z){(1+\overline{w}z)}^{-1}$ for the Möbius loop multiplication...