For more than 3 dimensions the angular velocity is best expressed as an antisymetric tensor ω^jn. ω^jn + ω^nj = 0. On a body rotating about the origin with angular velocity ω^jk, a point with coordinates x^j has velocity v^k = ω^jkx^j.

From first principles the angular momentum about the origin of a point mass dm located at x^j and with velocity v^k is dL_jk = (x^jv^k − x^kv^j)dm. The angular momentum of a flexible body is thus

L_jk =_∫ (x^jv^k − x^kv^j)dm.

For the rigid body, velocities stem from a uniform angular velocity ω^jk .
The angular momentum for the rigid body is thus
L_jk =_∫ (x^jω^nkxⁿ − x^kω^njxⁿ)dm = _∫ (x^j ω^nk xⁿ + x^k ω^jn xⁿ)dm
= _∫ (δ_vk x^j ω^iv xⁱ + δ_vn δ_ji x^k ω^iv xⁿ)dm = ω^iv _∫ (δ_vk x^j xⁱ + δ_vn δ_ji x^k xⁿ)dm
= ω^iv _∫ (δ_vk x^j xⁱ + δ_ji x^k x^v)dm = ω^iv (δ_{vk ∫} x^j xⁱ dm + δ_{ji ∫} x^k x^v dm)
= ω^iv (δ_vk m_ji + δ_ji m_kv) [= ω^iv (δ_vk δ_jα δ_iβ + δ_ji δ_vβ δ_kα) m_αβ]
= I_ivjk ω_iv
where I_ivjk = δ_vk m_ji + δ_ji m_kv.

Recall that both L_ij and ω_ij are antisymetric. Bypassing I_ivjk we may write

L_jk = I_ivjk ω_iv = (δ_vk m_ji + δ_ji m_kv) ω_iv = m_ji ω_ik + m_kv ω_jv.
L_ij = − m_iα ω_jα + m_jα ω_iα = ω_iα m_jα − ω_jα m_iα.

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.