The SO(3) group is the special orthogonal group in 3 (real) dimensions.

An example is orthogonal rotation matrices in R³.

Any such matrix B satisfies:

B^T⋅B=1 Orthogonality.

If, additionally, holds: \det B=1, then these describe pure rotation, otherwise mirror rotations which would change the handedness of the coordinate system would also be possible.

Group Axioms

The Group Axioms hold for SO(3):=(B∈{B: B^T⋅B=1}, ⋅):

  1. Closure: B_1^T⋅B_1=1, B_2^T⋅B_2=1 => (B_1⋅B_2)^T⋅(B_1⋅B_2)=B_2^T⋅B_1^T⋅B_1⋅B_2=1 You can't get out, even combinations hold.
  2. Associativity: (B_1⋅B_2)⋅B_3=B_1⋅(B_2⋅B_3)
  3. Neutral Element: B_1⋅1=B_1
  4. Inverse Element: B^-1⋅B=1 <=> B^-1=B^T

The Group is NOT an Abel Group, so there is NO commutativity.

Skew Symmetry of Special Cases

B^T⋅B=1 Orthogonality.
Ḃ^T⋅B+B^T⋅Ḃ=0
Ḃ^T⋅B=-B^T⋅Ḃ
Ḃ^T⋅B=-(Ḃ^T⋅B)^T Skew Symmetry.

It follows that there are only 3 (=3⋅(3-1)/2) linearly independent entries in B.

Vector to Matrix transformation

It follows that you can represent such a matrix B^T⋅Ḃ as a vector ω⃗:

ω⃗:=\begin{pmatrix} ω_1 \\ ω_2 \\ ω_3 \end{pmatrix}
B^T⋅Ḃ:=\begin{pmatrix} 0 & -ω_3 & ω_2 \\ ω_3 & 0 & -ω_1 \\ -ω_2 & ω_1 & 0 \end{pmatrix}

Hat Operator

Sometimes, the transformation between the vector and the matrix is written using the hat operator, not to be confused with Fourier transform hat operator or unit vector hat mark.

\hat{\vec{ω}}=B^T⋅Ḃ as above.

Then, the vector cross product and the matrix multiplication with one hatted vector have in common:

{\hat{\vec{ω}}}⋅r⃗=ω⃗⨯r⃗

And so:

B^T⋅Ḃ=\hat{\vec{ω}}
Ḃ=B⋅\hat{\vec{ω}}

Euler Rotations

The rotations themselves are NOT skew symmetric:

R_1(φ_1):=\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos φ_1 & -\sin φ_1 \\ 0 & \sin φ_1 & \cos φ_1 \end{pmatrix}
R_2(φ_2):=\begin{pmatrix} \cos φ_2 & 0 & \sin φ_2 \\ 0 & 1 & 0 \\ -\sin φ_2 & 0 & \cos φ_2 \end{pmatrix}
R_3(φ_3):=\begin{pmatrix} \cos φ_3 & -\sin φ_3 & 0 \\ \sin φ_3 & \cos φ_3 & 0 \\ 0 & 0 & 1\end{pmatrix}