  The goal of this note is to study the analog in unstable ${{\mathbb
A}^1}$-homotopy theory of the unit map from the motivic sphere spectrum to the
Hermitian K-theory spectrum, i.e., the degree map in Hermitian K-theory. We
show that "Suslin matrices", which are explicit maps from odd dimensional split
smooth affine quadrics to geometric models of the spaces appearing in Bott
periodicity in Hermitian K-theory, stabilize in a suitable sense to the unit
map. As applications, we deduce that $K^{MW}_i(F) = GW^i_i(F)$ for $i \leq 3$,
which can be thought of as an extension of Matsumoto's celebrated theorem
describing $K_2$ of a field. These results provide the first step in a program
aimed at computing the sheaf $\pi_{n}^{{\mathbb A}^1}({\mathbb A}^n \setminus
0)$ for $n \geq 4$.
