Final answer:
A matrix is skew-symmetric if it is equal to the negative of its transpose. The ordered basis for skew-symmetric matrices consists of matrices with the unique pattern of 1s and -1s off the diagonal. Given coordinates, a matrix in this space can be found by linearly combining the basis matrices with these coordinates as coefficients.
Step-by-step explanation:
A matrix is skew-symmetric if it is equal to the negative of its transpose, which means that for a skew-symmetric matrix A, we have AT = -A. Any square matrix can be represented as the sum of a symmetric and a skew-symmetric matrix. The elements on the diagonal of a skew-symmetric matrix must be zero because they are equal to their own negatives. Therefore, in a skew-symmetric matrix, only the entries above (or below) the diagonal need to be specified, as the others are just their negatives.
In a space of skew-symmetric matrices, the ordered collection with matrices that have a 1 in a unique position above the diagonal and 0s elsewhere, with the corresponding symmetric entry below the diagonal being -1, forms an ordered basis. For example, in three dimensions, the basis for the space of 3x3 skew-symmetric matrices would consist of matrices with a single 1 and -1 in symmetric positions off the diagonal, and 0 elsewhere.
If a matrix in this space has certain coordinates, you can construct it by taking a linear combination of the basis matrices, using the given coordinates as coefficients. For instance, if the matrix has coordinates (a, b, c), it would be a combination of the three basis matrices, each multiplied by a, b, or c, respectively.