Manddelberg [9] has shown that a Clifford algebra of a free quadratic space over an arbitrary semi-local ring R in Brawer-Wall group BW(R) is determined by its rank, determinant, and Hasse invariant. In this paper, we prove a corresponding result when R is a full ring.Throughout this paper, unless otherwise specified, we assume that R is a commutative ring having 2 a unit. A quadratic space (V, B, .phi.) over R is a finitely generated projective R-module V with a symmetric bilinear mapping B: V*V.rarw.R which is non-degenerate (i.e., the natural mapping V.rarw.Ho

(V,R) induced by B is an isomorphism), and with a quadratic mapping .phi.: V.rarw.R such that B(x,y)=1/2(.phi.(x+y)-.phi.(x)-.phi.(y)) and .phi.(rx) =

.phi.(x) for all x, y in V and r in R. We denote the group of multiplicative units of R by U9R). If (V, B, .phi.) is a free rank n quadratic space over R with an orthogonal basis {

,..,

}, we will write <

,..,

> for (V, B, .phi.) where the

=.phi.(

) are in U(R), and denote the space by the table [

] where

=B(

,

). In the case n=2 and B(

,

)=1/2 we reserve the notation [a

,

] for the space. A commutative ring R having 2 a unit is called full [10] if for every triple

,

,

of elements in R with (

,

,

)=R, there is an element w in R such that

+

w+

=unit.TEX>=unit.t.t.t.