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or in component form, if A = (a_{i,j}):
For example, the following matrix is skewsymmetric:
All main diagonal entries of a skewsymmetric matrix have to be zero, and so the trace is zero.
The skewsymmetric nbyn matrices form a vector space of dimension (n^{2}  n)/2. This is the tangent space to the orthogonal group O(n). In a sense, then, skewsymmetric matrices can be thought of as "infinitesimal rotations".
In fact, the skewsymmetric nbyn matrices form a Lie algebra using the commutator Lie bracket
A matrix G is orthogonal and has determinant 1, i.e., it is a member of that connected component of the orthogonal group in which the identity element lies, precisely if for some skewsymmetric matrix A we have
See also symmetric matrix.
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