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For more than 3 dimensions the angular velocity is best expressed as an antisymetric tensor ωjn. ωjn + ωnj = 0. On a body rotating about the origin with angular velocity ωjk, a point with coordinates xj has velocity vk = ωjkxj.

From first principles the angular momentum about the origin of a point mass dm located at xj and with velocity vk is dLjk = (xjvk – xkvj)dm. The angular momentum of a flexible body is thus

Ljk = (xjvk – xkvj)dm.

For the rigid body, velocities stem from a uniform angular velocity ωjk .
The angular momentum for the rigid body is thus
Ljk = (xjωnkxn – xkωnjxn)dm = (xj ωnk xn + xk ωjn xn)dm
= vk xj ωiv xi + δvn δji xk ωiv xn)dm = ωiv vk xj xi + δvn δji xk xn)dm
= ωiv vk xj xi + δji xk xv)dm = ωivvk xj xi dm + δji xk xv dm)
= ωivvk mji + δji mkv) [= ωivvk δ δ + δji δ δ) mαβ]
= Iivjk ωiv
where Iivjk = δvk mji + δji mkv.

Recall that both Lij and ωij are antisymetric. Bypassing Iivjk we may write

Ljk = Iivjk ωiv = (δvk mji + δji mkv) ωiv = mji ωik + mkv ωjv.
Lij = – m ω + m ω = ω m – ω m.

In either form L has been expressed as a bilinear form in the second moments m and angular velocity ω. It is also manifestly covariant considering the orthogonal coördinates.

This program corroborates these derivations.