Show that C^x is isomorphic to GL2(R)

Show that C^x is isomorphic to the subgroup of GL2(R) consisting of all matrices of the form:

a b

-b a such that a^2+b^2 is not 0.

I'm guessing I need to show C^x--->G given by c(a+bi)=

a b

-b a is an isomorphism.

I need to show 1-1, onto, and homomorphism.

1-1:

Assume c(a+bi)=c(c+di). I need to show this implies a+bi=c+di

a b c d

-b a = -d a

I'm not sure how this implies 1-1

onto:Not sure how to show this

homomorphism:

c(ab)=c(a)c(b)

Let a=a+bi, b=c+di

a b c d

-b a * -d c

ac-db ad+bc

-bc-ad -bd+ac =c(ab)