We know how annoying are the HTML products generated by any of the MS Office applications.
What should be a simple and clean table ends to be a heavy document full of unnecessary tags that make the most of the HTML output.
From Excel we generally just need clean tables, to do so just use Excel itself.
Look for some unused portion of the spreadsheet or use a new empty one, and just write a formula for each column you want in HTML.
Let's suppose your columns are A, B, C and the data start at row 5.
Just write the following formulas in your empty space:

• The first column ="<tr><td>" & A5 & "</td>"
• The second column ="<td>" & B5 & "</td>"
• The third column ="<td>" & C5 & "</td></tr>"

Note: you can customize your inserted HTML in the formulas with any attributes and extra tags you want to enhance the presentation adding colors, changing fonts, etc.

Then just extend the 3 rows file down to cover all the rows with actual data.
After that you will have all your cells populated with actual clean HTML code.
Select only those cells, copy them to the clipboard and insert them in your HTML document between the <table> and </table> tags.