A spinor representation for the conformal group of the real orthogonal space
R
p
,
q
{R^{p,q}}
is given. First, the real orthogonal space
R
p
,
q
{R^{p,q}}
is compactified by adjoining a (closed) isotropic cone at infinity. Then the nonlinear conformal transformations are linearized by regarding the conformal group as a factor group of a larger orthogonal group. Finally, the spin covering group of this larger orthogonal group is realized in the Clifford algebra
R
1
+
p
,
q
{R_{1 + p,q}}
containing the Clifford algebra
R
p
,
q
{R_{p,q}}
on the orthogonal space
R
p
,
q
{R^{p,q}}
. Explicit formulas for orthogonal transformations, translations, dilatations and special conformal transformations are given in Clifford language.